Logic Seminar : Past Events
Past Seminars
The following seminars have already happened, you may instead view upcoming seminars in this series.
Aug. 23, 2005
Chris Miller :
4 p.m. in SEO 512
Abstract
Let $\mathfrak R$ be an o-minimal expansion of $(\mathbb R,<,+)$ and $(\phi_k)_{k\in\mathbb N}$ be a
sequence of positive real numbers such that $\lim_{k\to+\infty}f(\phi_k)/\phi_{k+1}=0$ for every $f
\colon\mathbb R\to \mathbb R$ definable in $\mathfrak R$. (Such sequences always exist under some
reasonable extra assumptions on $\mathfrak R$, in particular, if $\mathfrak R$ is exponentially bounded
or if the language is countable.) Then $\bigl(\mathfrak R, (S)\bigr)$ is d-minimal, where $S$ ranges over
all subsets of cartesian powers of the range of $\phi$. (Joint work with Harvey Friedman. Published in JSL
70.)
Aug. 30, 2005
John Baldwin :
4 p.m. in SEO 427
Abstract
We will consider the relation between abstract proofs of `tameness'
and Zilber's study of covers of semi-abelian varieties.
Sept. 6, 2005
Greg Hjorth :
4 p.m. in SEO 427
Abstract
An equivalence relation $E$ on $X$ is treeable if there is an acyclic graph $R$ which is Borel as a subset of
$X\times X$ and whose connected components form the $E$-equivalence classes.
I will survey what is known about treeable equivalence relations up to Borel reducibility and orbit
equivalence. I will also mention a dichotomy theorem for when a treeable equivalence relation is reducible
to one with countable classes and another dichotomy theorem for when it is possible to select an end
from each equivalence class.
Sept. 15, 2005
John Baldwin :
4 p.m. in SEO 427
Abstract
This is a continuation of the talk on August 30th. We will consider the relation between abstract proofs of
`tameness' and Zilber's study of covers of semi-abelian varieties.
Sept. 20, 2005
Dave Marker :
4 p.m. in SEO 427
Abstract
We will look at Zilber's axioms for pseudoexponentiation. In particular we will wonder if the simplest case
of the strong exponential closure axiom is true for the complex exponential.
Oct. 6, 2005
Mikhail Kotchetov :
4 p.m. in SEO 427
Abstract
Berarducci (2000) studied irreducible elements of the ring $k((G^{<0}))\oplus \Z$, which is an integer
part of the power series field $k((G))$ where $G$ is an ordered divisible abelian group and $k$ is an
ordered field. Pitteloud (2001) proved that some of the irreducible elements constructed by Berarducci
are actually prime. Both authors mainly concentrated on the case of archimedean $G$. In this paper, we
study {\it truncation integer parts} of any real closed field and generalize results of Berarducci and
Pitteloud. In particular, we prove that $k((G^{<0}))\oplus\Z$ has (cofinally many) prime elements for
any ordered divisible abelian group $G$. Addressing a question in the paper of Berarducci, we show
that every truncation integer part of a non-archimedean exponential field has a cofinal set of
irreducible elements. (Joint with S. Kuhlmann and D. Biljakovic.)
Oct. 11, 2005
Kirsten Eisenträger :
4 p.m. in SEO 427
Abstract
Hilbert's Tenth Problem in its original form was to find an algorithm to decide, given a polynomial
equation $f(x_1,\dots,x_n)=0$ with coefficients in the ring $\mathbf{Z}$ of integers, whether it has a
solution with $x_1,\dots,x_n \in \mathbf{Z}$. Matiyasevich proved that no such algorithm exists, i.e.
Hilbert's Tenth Problem is undecidable. Since then, analogues of this problem have been studied by
asking the same question for polynomial equations with coefficients and solutions in other
commutative rings.
Let $k$ be the function field of a curve over a finite field, and let $v$ be a non-trivial discrete valuation
on $k$ with valuation ring $R_v$. We will give a new proof of the known result that $R_v$ is
diophantine over $k$, and we will show how this can be used to prove that Hilbert's Tenth Problem for
$k$ is undecidable.
Oct. 18, 2005
Anand Pillay :
4 p.m. in SEO 427
Abstract
Influenced by work of Chatzidakis-Hrushovski asnd using results of Bost, we show that the Grothendieck-
Katz conjecture is equivalent to a certain nonlinear generalization (introduced by Ekedahl et al) modulo a
differential-algebraic statement concerning "nonorthogonality to the constants".
Oct. 25, 2005
Ozlem Beyarslan :
4 p.m. in SEO 427
Abstract
A pseudofinite field is an infinite field satisfying all first-order properties which hold in all finite fields.
Pseudofinite fields exist and they can be realized, for example, as ultraproducts of finite fields.
An $n$-ary random graph is a set $X$ with a symmetric and irreflexive $n$-ary relation $R$ such that
for any two finite and disjoint subsets $A$ and $B$ of $X^{n-1}$, there is an $x\in X$ such that $R(a,x)
$ and $\neg R(b,x)$ for all $a\in A$ and $b\in B$.
In 1980 J. L. Duret interpreted a random binary graph in a pseudofinite field. This has some important
model theoretic consequences.
We will show that we can interpret a random $n$-ary graph in pseudofinite fields.
Nov. 1, 2005
Beibut Kulpeshov :
4 p.m. in SEO 427
Abstract
We explore analogues of o-minimality and weak o-minimality for circularly ordered sets. Much of the
theory goes through almost unchanged, since over a parameter the circular order yields a definable linear
order. Working over $\emptyset$ there are differences. Our main result is a structure theory for $\aleph_0
$-categorical weakly circularly minimal structures. This is joint work with Dugald Macpherson.
Nov. 8, 2005
Antonio Montalbán :
4 p.m. in SEO 427
Abstract
We analyze the structure of equimorphism types of linear orderings ordered by embeddability. (Two
linear orderings are equimorphic if they can be embedded in each other.) Our analysis is mainly from
the viewpoints of Computable Mathematics and Reverse Mathematics. But we also obtain results, as the
definition of equimorphism invariants for linear orderings, which provide a better understanding of the
shape of this structure in general.
Here are our main results: Spector proved in 1955 that every hyperarithmetic ordinal is isomorphic to a
computable one. We extend his result and prove that every hyperarithmetic linear ordering is
equimorphic to a computable one. From the viewpoint of Reverse Mathematics, we look at the strength
of Fraïssé's conjecture. From our results, we deduce that Fraïssé's conjecture is sufficient and
necessary to develop a reasonable theory of equimorphism types of linear orderings.
Nov. 10, 2005
Andrew Coppola :
2 p.m. in SEO 427
Abstract
We consider theories in logics equipped with the generalized quantifier that asks, "do there exist |M|-
many". The goal is to study these within the framework of Abstract Elementary Classes. However, they
are slightly more general as they do not satisfy the full axiomatic chain conditions on substructures. One
target is a variant of the Los Conjecture for these classes. This reconciles results approached from very
different directions. Lawskoski and Pillay prove the Los Conjecture for the class of Gross Models (models
of a complete first-order theory where all infinite definable sets have cardinality equal to the model). This
can be recast in the work on tame AECs, initiated by Grossberg and Vandieren.
Nov. 22, 2005
Su Gao :
4 p.m. in SEO 427
Abstract
We consider quotient Boolean algebras of the form P(N)/I, where I is an ideal on N, and their isomorphism
types. Mike Oliver in his PhD dissertation proved that there are continuum many non-isomorphic such
Boolean algebras induced by Borel ideals. In a joint work with Oliver we show that every analytic
equivalence relation is Borel reducible to this isomorphism relation.
Nov. 29, 2005
Beibut Kulpeshov :
4 p.m. in SEO 427
Abstract
This talk concerns the notion of weak o-minimality originally and deeply studied by D. Macpherson, D.
Marker and C. Steinhorn [TAMS, 2000]. Real closed fields with a proper convex valuation ring provide
an important example of weakly o-minimal structures. A. Pillay and C. Steinhorn have described all $
\aleph_0$--categorical o-minimal theories [TAMS, 1986]. Their description implies binarity for these
theories. Here we present some results on $\aleph_0$--categorical weakly o-minimal theories, and
discuss some connections between two notions: binarity and convexity rank. Recall that convexity rank
for a formula with one free variable was introduced by the speaker in [JSL, 1998]. In particular, a theory
has convexity rank 1 if there is no definable (with parameters) equivalence relation with infinitely many
infinite convex classes. It is obvious an o-minimal theory has convexity rank 1. Firstly, we give a
description of $\aleph_0$--categorical binary weakly o-minimal theories of convexity rank 1 [A & L,
2005]. Further, we present some technique on 2--formulas which was originally introduced by B.S.
Baizhanov. At last, by using this technique we obtain a criterion for binarity of $\aleph_0$--categorical
weakly o-minimal theories in terms of convexity rank (the main result of the talk).
Jan. 10, 2006
Dave Marker :
4 p.m. in SEO 427
Abstract
Greg Hjorth proved that if there is a counterexample to Vaught's Conjecture, then there is one with no
model of size aleph_2. The proof is an interesting mix of model theory and descriptive set theory. I will
describe the proof and some of the necessary background material.
Jan. 17, 2006
Dave Marker :
4 p.m. in SEO 427
Abstract
Greg Hjorth proved that if there is a counterexample to Vaught's Conjecture, then there is one with no
model of size aleph_2. The proof is an interesting mix of model theory and descriptive set theory. I will
describe the proof and some of the necessary background material.
This is a continuation of last week's talk.
Jan. 24, 2006
John Baldwin :
4 p.m. in SEO 427
Abstract
We will discuss the results of Makkai and Harnik that every counterexample to the Vaught conjecture has
an uncountable model; indeed it has both an uncountable model which is $\infty,\omega$-equivalent to a
countable model and one which is not. We give an `admissible set free' proof of the first result. Further,
we observe that any first order counterexample to Vaught's conjecture has $2^{\aleph_1}$ models of
power $\aleph_1$.
Jan. 31, 2006
Sergio Fratarcangeli :
3 p.m. in SEO 427
Abstract
Khovanskii Theory is an important tool for identifying new o-minimal structures and for obtaining
uniformity results within o-minimal structures. In this talk, we generalize Khovanskii Theory in two
directions. In one case, we produce a version that holds within any expansion of an ordered field with the
intermediate value property. In the other case, we generalize how sets may be obtained from Rolle
leaves---beyond mere intersections---so that the number of their connected components is still
uniformly bounded.
Feb. 7, 2006
Chris Miller :
3 p.m. in SEO 427
Abstract
I will discuss the following result, which is part of a joint project with A. Dolich and C. Steinhorn.
Theorem. Let T be a complete theory extending the theory of densely ordered groups such that, in every
model of T, every unary open definable set is a finite union of open intervals. Then, for every model of T,
the reduct generated by its open definable sets (of all arites) is o-minimal.
Feb. 14, 2006
John Baldwin :
4 p.m. in SEO 427
Abstract
I will discuss approaches to Vaught's Conjecture by counting the number of models with cardinality $
\aleph_1$. The main actual result will be a theorem of Gao: An infinite structure $M$ is totally
categorical if and only if the automorphism group of $M$ admits a complete left-invariant metric. We will
observe that that fact prevents naive attempts to construct sentences of $\mathcal L_{\omega_1,\omega}$
with a prescribed number of models of cardinality $\aleph_1$.
Feb. 21, 2006
Andrew Coppola :
4 p.m. in SEO 427
Abstract
We will discuss a surprising fact about these sequences of natural numbers which appear to grow at an
explosive rate, and sketch a proof about them using perhaps equally surprising techniques from logic.
Feb. 28, 2006
Dave Marker :
4 p.m. in SEO 427
Abstract
Consider the theory of the natural numbers where we use the quantifiers "almost all" and "exists
unboundedly many" instead of "for all" and "there exists". Ted Slaman proved the surprising result that
this theory is decidable. We will show that this result has an easy model theoretic proof.
March 7, 2006
Jana Maríková :
4 p.m. in SEO 427
Abstract
We let $M$ be a big o-minimal structure and say that a group is a $\iota$-group if the underlying set and
the graph of the group operation are automorphism invariant subsets of $M^n$ and $M^3n$ respectively.
We show that a $\iota$-group $G$ in $M^n$ has a unique topology making it a topological group and
inducing the same topology on a large $\iota$-subset of $G$ as $M^n$. If $G$ is in particular a type-
definable group in $M^n$ then the group topology on $G$ is induced by a definable manifold. This
implies in the case when $M$ is an o-minimal expansion of a real closed field that $G$ is affine.
March 16, 2006
David Ross :
4 p.m. in SEO 427
Abstract
Almost halfway through its third millennium, nonstandard measure theory is now a relatively mature
subject. I'll give a quick introduction to the subject as it exists today, with some recent examples of
nonstandard measure representations and constructions.
March 28, 2006
Elisabeth Bouscaren :
4 p.m. in SEO 427
Abstract
The existence of a finitely axiomatisable $\aleph_1$-categorical theory with non-trivial geometry is open.
Hrushovski has shown that such a theory must have a locally modular geometry, and conjectured that the
existence of such a theory was equivalent to the existence of a division ring finitely presented as a ring. In
joint work with Thomas Blossier, we have shown that if $G$ is a finitely axiomatisable strongly minimal
group, then the ring of quasi-endomorphisms of G is indeed finitely presented.
April 3, 2006
Thomas Kucera :
4 p.m. in SEO 427
Abstract
The elementary socle of a module is the sum of all the minimal non-zero first-order definable
subgroups of that module. Dually the elementary radical of a module is the intersection of all the
maximal proper first-order definable subgroups of that module. These concepts were first introduced
by Ivo Herzog in his thesis.
If an indecomposble module has the descending chain condition on definable subgroups, the
elementary socle is non trivial and is a definably closed submodule. Furthermore, the definition of
elementary socle naturally extends to an ascending series of definably closed submodules whose union
is the whole module. Dually, if an indecomposable module is pure-injective and has the ascending
chain condition on definable subgroups, the elementary radical is a submodule, and the definition of
the elementary radical may be extended to a descending series of submodules whose intersection is 0.
The definitions and some of the properties generalize in natural ways to arbitrary (indecomposable)
pure-injective modules.
Mike Prest introduced a notion of duality between certain first order formulas in the languages of left
modules and right modules which Herzog extended to a duality of categories. This duality makes
modules of the kind described above correspond; and their internal structure is shown to be similar by
means of this duality.
This is a preliminary report on work in progress.
April 4, 2006
Alex Wilkie :
4 p.m. in SEO 427
Abstract
Every closed subset of euclidean space can be represented as the zero set of a smooth function (ie an
infinitely differentiable function defined on the whole ambient space). My student, Gareth Jones, has
been investigating the question of whether this theorem (of Whitney) holds in the context of sets and
functions definable in an o-minimal expansion of the real exponential field. This turns out to require
uniform estimates on all derivatives of a given definable function. In many structures of interest these
estimates are easily seen to hold for the terms of the language, so the problem becomes one of finding
analytic operations, mapping definable functions to definable functions, which (a) preserve the
estimates and (b) eventually generate all definable functions. This is the topic of my talk.
April 10, 2006
Chris Laskowski :
4 p.m. in SEO 427
Abstract
Shelah and Spencer proved that when alpha is irrational in (0,1) there is an almost sure theory T of
random graphs of size n, when the edge probability is n^{-alpha}. Somewhat later, Baldwin observed
that the same theory T can be visualized as the theory of a generic object of a generalized Fraisse
construction.
We give an AE axiomatization of T and examine other countable models of T. Somewhat surprisingly,
even though it can be uniquely characterized as being a Fraisse limit, the generic does not appear to be
any particularly `special' model of T.
We will also discuss applicability of these methods to other Fraisse constructions.
April 11, 2006
Patrick Speissegger :
4 p.m. in SEO 427
Abstract
Let f be a totally defined real-valued function on the real line. Borel's Lemma states that if f is increasing
and everywhere greater than or equal to 1, and if r>1 is fixed, then the set {x: f(x + 1/f(x)) >= r f(x)} has
outer measure at most r/(r-1). Recently, Chris Miller conjectured that a suitably restated version of Borel's
Lemma was true in the o-minimal setting. Interestingly, the proof of this version appears to be more
elementary for exponential o-minimal structures than for power-bounded ones. (Joint work with Alf
Dolich and Chris Miller)
April 18, 2006
Krzysztof Krupinski :
4 p.m. in SEO 427
Abstract
A profinite structure in the sense of Newelski is a pair $(X,Aut^*(X))$ consisting of a profinite
topological space $X$ and a closed subgroup $Aut^*(X)$ (called the structural group) of the group of
all homeomorphisms of $X$ respecting the inverse system defining $X$. We say that a profinite
structure $(X,Aut^*(X))$ is small if for every natural number $n>0$, there are only countably many
orbits on $X^n$ under the action of the structural group. In small profinite structures Newelski
introduced a topological notion of independence, which has similar properties to those of forking
independence in stable theories, and developed a counterpart of geometric stability theory in this
context.
I will present this notion of independence and explain why smallness plays an important role here. I will
also give some examples and results concerning small profinite groups regarded as profinite
structures.
Then I will talk about my recent ideas concerning generalizations of small profinite structures to the
case of: 1) non-small profinite structures; 2) 'compact structures' (i.e. $X$ is a compact metric space
and $Aut^*(X)$ is a compact group acting on $X$ continuously); 3) 'Polish structures' (i.e. $X$ is a
Polish space and $Aut^*(X)$ is a Polish group acting on $X$ continuously).
April 25, 2006
Charles Steinhorn :
4 p.m. in SEO 427
Abstract
We discuss joint work with A. Onshuus concerning notion(s) of rank for linearly ordered structures. Ever
since o-minimal theories were defined, it has been thought that they might play a role with respect to
linearly ordered structures analogous to that played by strongly minimal theories in the stable context.
The talk focuses on the authors' efforts to begin to develop this analogy via the introduction of rank(s) for
ordered structures.
April 27, 2006
Monica van Dieren :
2 p.m. in SEO 427
Abstract
We will discuss limit and super-limit models in the context of both abstract elementary classes and first
order model theory.
April 28, 2006
Amador Martín-Pizarro :
11 a.m. in SEO 427
Abstract
In this talk I will present joint work with A. Baudisch, M. Hils and F. Wagner on Poizat's green fields. These
fields are an infinite rank candidate for a bad field in characteristic 0. We will discuss how a result on
Schanuel-like inequalities due to Ax allowed Poizat to obtain a finiteness result on the toric equations to
consider for expressing Q-linear independence for certain extensions in the Fraisse-Hrushovski model.
We will conclude from this the existence of a collapsed structure, i.e., a field of rank 2 with a definable
divisible proper multiplicative subgroup.
May 9, 2006
Andrés Villaveces :
4 p.m. in SEO 427
Abstract
I will present a new Fraïssé-Hrushovski style construction within the framework of continuous model
theory, whose net result is a limit model of the theory of Hilbert Spaces with a generic distance function
(to a generic "black spot"). This provides a new application of construction of limit models and a new
example of a simple theory - in continuous model theory. This is joint work with Alexander Berenstein.
Aug. 29, 2006
Chris Miller :
4 p.m. in SEO 427
Abstract
Let $F\colon \mathbb R^n\to \mathbb R^n$ be linear and $\gamma\colon \mathbb R\to \mathbb R^n$
be differentiable such that $\gamma'(t)=F(\gamma(t))$ for all $t\in \mathbb R$. Then the image $
\gamma(\mathbb R)$ is interdefinable over the real field with at least one of: the real exponential function
$e^x$; the complex exponential function $e^z$; or a finite set of functions $t\mapsto t^w\colon (0,\infty)
\to \mathbb C$, where $w\in\mathbb C$. Moreover, which case(s) hold can be semialgebraically
computed from the coefficients of $F$.
Sept. 5, 2006
Dave Marker :
4 p.m. in SEO 427
Abstract
If E is a Borel equivalence relation with countably many classes, then there is an infinitary sentence $\phi$
such that E is Borel bi-reducible to the isomorphism relation for $\phi$. Kechris and Hjorth asked if the
same was true for first order theories. We show that if a theory has uncountably many types, then the
isomorphism relation is more complicated than any Borel equivalence relation with countably many
classes.
Sept. 12, 2006
Salih Azgin :
4 p.m. in SEO 427
Abstract
A classical result by Ax-Kochen and Ershov provides a very good understanding of the model theory of
finitely ramified henselian valued fields. ``Model Theory of Frobenius on Witt Vectors'' (preprint, Belair-
Macintyre-Scanlon) pursues a similar goal in the context of valued difference fields, i.e., valued fields
with a distinguished automorphism.
Our goal is to prove an analogue of the main result of the BMS paper in equal characteristic $p>0$. We
shall start by considering valued difference fields in equal characteristic zero, where the interaction
between the valuation and the distinguished automorphism is different than that of the BMS context.
Even though this seems irrelevent to the question at hand, it provides good insight to the model theory
valued fields and valued difference fields in equal characteristic $p>0$.
Sept. 19, 2006
Dave Marker :
4 p.m. in SEO 427
Abstract
If E is a Borel equivalence relation with countably many classes, then there is an infinitary sentence $\phi$
such that E is Borel bi-reducible to the isomorphism relation for $\phi$. Kechris and Hjorth asked if the
same was true for first order theories. We show that if a theory has uncountably many types, then the
isomorphism relation is more complicated than any Borel equivalence relation with countably many
classes. This is a continuation of the talk from September 5.
Sept. 26, 2006
Alf Dolich :
4 p.m. in SEO 427
Abstract
Recall that a theory T is said to satisfy uniform finiteness if for any uniformly definable family
of sets there is a natural number N so that any finite set in the family has cardinality at most N.
Let T be an expansion of the theory of dense linear orderings. If T does not satisfy uniform finiteness
then in some model of T there is an infinite definable discrete set. We consider consequences of this fact
and investigate when the converse holds and examples where it does not.
Oct. 3, 2006
Kathryn Vozoris :
4 p.m. in SEO 427
Abstract
We will discuss model theoretic properties of the complex field with a predicate for the integers. In
particular that the theory is quasiminimal, has quantifier elimination up to quantification over the integers
and is model complete. We will also consider some definability results.
Oct. 10, 2006
Patrick Speissegger :
4 p.m. in SEO 427
Abstract
I start by outlining one of the strategies used to establish the o-minimality of certain expansions of the
real field. Working with Tobias Kaiser and Jean-Philippe Rolin, we are now trying to use apply this strategy
to the correspondence maps of polycycles of planar analytic vector fields, in the case where all the
singularities along the polycycle are hyperbolic. It turns out, based on Dulac's and Ilyashenko's work, that
if one assumes in addition that these hyperbolic singularities are all non-resonant, then a direct
application of one of the established techniques works. I will explain this result and speculate on its
possible extension to the general hyperbolic case.
Oct. 17, 2006
Jonathan Kirby :
4 p.m. in SEO 427
Abstract
The complex field with exponentiation is a structure about which surprisingly little is known. Zilber
conjectured that it is isomorphic to the "pseudoexponentiation" he constructed using Hrushovski's
amalgamation technique. This conjecture is very hard, because it implies Schanuel's conjecture of
transcendental number theory and would give a complete description of the algebraic relations between
exponentials.
However, Zilber's conjecture can be separated into this difficult number-theoretic part and a geometric
part. I will describe how the geometric part is related to the model theory of differential fields, and
will outline the progress towards proving it.
Oct. 24, 2006
Kathryn Vozoris :
4 p.m. in SEO 427
Abstract
I will discuss the notion of Morley rank modulo a predicate (PMR) recently introduced by Jacob
Heidenreich. This is a generalization of Morley rank designed for structures with a predicate, where the
underlying structure without the predicate is totally transcendental. The aim of PMR is to measure the
dimension of a structure without taking into account the definable sets generated by the predicate. I will
discuss the proof that the PMR of the complex field with a predicate for the integers is larger than every
natural number.
Oct. 31, 2006
Jonathan Kirby :
4 p.m. in SEO 427
Abstract
I will give a complete axiomatization of the theory of exponential differential equations in the context
of differential fields.
The axiomatization consists of a description of which systems of equations can and cannot have
solutions. It builds on James Ax's differential field version of Schanuel's conjecture of transcendental
number theory. The method works for the equations satisfied by the exponential maps of any
semiabelian variety, but in this talk I will concentrate mainly on the usual exponentiation.
The first-order nature of the axiomatization is closely related to an application in diophantine
geometry, which will be discussed more carefully in a future number theory seminar. An application of
this work to complex exponentiation was discussed at a previous seminar.
Nov. 7, 2006
Matthias Aschenbrenner :
4 p.m. in SEO 427
Abstract
I will tell you what I know about the topic indicated in the title.
Nov. 21, 2006
Pantelis Eleftheriou :
4 p.m. in SEO 427
Abstract
We discuss Pillay's Conjecture (PC) and Compact Domination Conjecture (CDC) for groups definable in
linear o-minimal structures. Let $G$ be a definably compact group definable in a saturated o-minimal
structure $\mathcal{M}$. Roughly stated, PC says that $G$ must contain a normal type-definable
subgroup $G^{00}$ of `infinitesimals', such that $G/G^{00}$ is a real compact Lie group of the same
dimension as $G$. CDC says that, in this case, the canonical homomorphism $\pi:G\rightarrow G/G^
{00}$ is a kind of intrinsic `standard part map'.
If $\mathcal{M}$ expands a real closed field, then PC is true (Hrushovski-Peterzil-Pillay) and CDC
remains open.
If $\mathcal{M}$ is an ordered vector space over an ordered division ring, we first prove that $G$ is a
`definable torus', and then answer positively both PC and CDC.
Nov. 28, 2006
Matthias Aschenbrenner :
4 p.m. in SEO 427
Abstract
This will be a continuation of my talk on November 7.
Dec. 5, 2006
Gareth Jones :
4 p.m. in SEO 427
Abstract
Theorem 5.1 in Wilkie's exponentiation paper gives a method for constructing points on varieties defined
by certain smooth definable functions. I will generalize this to to locally polynomially bounded structures.
These leads to a description of the definable functions in such structures. This is joint work with Alex
Wilkie.
Jan. 23, 2007
John Baldwin :
3 p.m. in SEO 427
Abstract
By the weak GCH we mean the assertion: 2^\lambda < 2^{\lambda^+}. We will
argue for the acceptance of this axiom on grounds of its coherence with our
intuitions of cardinal arithmetic and its consequences for mathematical
practice. The second argument is mediated by the Devlin-Shelah weak diamond. We
will explain this principle, which follows easily from weak GCH, and discuss
some of its consequences. These include a version of Morley\'s theorem for
L_{\omega_1,\omega} (Shelah) and implications for the study of semi-abelian
varieties (Zilber).
<p>
The goal of the talk will be to derive Weak Diamond from weak GCH, sketch a bit
of context/consequences, and pose some open problems.
Jan. 30, 2007
Jonathan Kirby :
3 p.m. in SEO 427
Abstract
In this basic talk I will explain what pregeometries are and why they are
useful. I will give a few definitions of pregeometries for real, complex and
abstract exponentiation, mostly introduced by Alex Wilkie, and will discuss a
couple of open problems about them.
April 3, 2007
Bill Howard :
3 p.m. in SEO 427
Abstract
An ordinal is said to be predicative if it measures the strength of a
predicative formal system of analysis (ie., a system based on predicative
definitions of sets of natural numbers and proofs). Kreisel, Schuette and
Feferman proposed in the 1960s that a certain ordinal (Gamma_0) is the least
upper bound of the predicative ordinals. This is open to controversy (see FOM
last spring: Weaver vs. others). I shall examine the situation from the
viewpoint of the theory of constructions.
Aug. 28, 2007
David Marker :
4 p.m. in SEO 427
Abstract
In 1959 Julia Robinson showed that the ring of integers is definable in the
field of rational numbers. I will give a more conceptual proof due to Rumley.
This result is at the core of Scanlon's recent work on definabilty in finitely
generated fields.
Sept. 4, 2007
David Marker :
4 p.m. in SEO 427
Abstract
A contiuation of my August 28 seminar.
Sept. 11, 2007
Chris Miller :
4 p.m. in SEO 427
Abstract
Let f be an entire function of one complex variable, regarded as a map from R^2
to R^2. It is known that the expansion of the real field (R,+,x) by all
restrictions of f to compact balls is o-minimal, and similarly for all
restrictions of the maximum function M_f(r):=max{|f(z)|:|z|=r}, r>0.
Preliminary investigation suggests that, under any "reasonable" assumptions,
(R,+,x,f) defines the set of all integers. On the other hand, the situation
regarding M_f is quite unclear. In particular, I have no examples where I know
that (R,+,x,M_f) is not o-minimal (though surely some must exist). I will
illustrate the issues by examining the case that f is defined by an Euler
partition product.
Sept. 18, 2007
Anand Pillay :
4 p.m. in SEO 427
Abstract
We formulate and prove an analogue of the Lindemann-Weierstrass
theorem for semiabelian varieties over function fields. (Joint work
with D. Bertrand.)
Sept. 20, 2007
Martin Koerwien :
4 p.m. in SEO 427
Abstract
We compare two different notions of complexity: depth of a classifiable theory
(in the sense of S. Shelah's Classification Theory) on the one hand side and
Borel reducibility as defined by H. Friedman and L.Stanley on the other hand
side. We show how these notions are related to the classification problem for
classes of countable models of a theory and give some positive and negative
results concerning their mutual relationship.
Oct. 2, 2007
Alexey Ovchinnikov :
4 p.m. in SEO 427
Abstract
Having a system of algebraic partial or ordinary differential equations one
can answer the questions like whether the system is inconsistent
or not or whether another equation is a consequence
of the system. This can be done by means of differential
elimination. In the talk we will look at the basic
notions of differential algebra and at a differential
elimination algorithm.
Oct. 9, 2007
Alexey Ovchinnikov :
4 p.m. in SEO 427
Abstract
Continuing the previous talk we shall look
at properties of the differential elimination algorithm.
In particular, in the ordinary case we shall bound orders
of derivatives that occur in the algorithm and discuss
effective differential Nullstellensatz.
Oct. 16, 2007
Alice Medvedev :
4 p.m. in SEO 427
Abstract
A difference field is a field with a distinguished automorphism
$\sigma$; for example, one can look at a field of characteristic $p$
and take $\sigma(x) = x^p$ to be the Frobenius automorphism.
Model-theorists have things to say about the class of
existentially-closed difference fields: it is first-order
axiomatizable, and the resulting theory, called ACFA, is supersimple
and generally nice. I will say things certain minimal definable sets
in ACFA, namely about solution sets of $\sigma(x) = f(x)$ for rational
functions $f$.
Oct. 23, 2007
Alice Medvedev :
4 p.m. in SEO 427
Abstract
A continuation of the October 16 seminar.
Oct. 30, 2007
David Marker :
4 p.m. in SEO 427
Abstract
Pillay answered a long-standing open problem of Kolchin's by proving that any differential algebraic group can be
differentially embedded into an algebraic group. I will discuss a simplification of the original proof due to Kowalski and Pillay.
Nov. 6, 2007
Isaac Goldbring :
4 p.m. in SEO 427
Abstract
Hilbert's Fifth Problem for Local Groups roughly asks whether every locally euclidean local group
is locally isomorphic to a Lie group. After giving a brief history, I will present a positive solution to
the question. The proof uses nonstandard methods and the relevant notions from nonstandard analysis will be
explained.
Nov. 13, 2007
Meeri Kesala :
4 p.m. in SEO 427
Abstract
Finitary abstract elementary classes is a subclass of AECs which
was motivated by some well-behaved syntactical examples of AECs such as
excellent sentences in $L_{\omega_1,\omega}$ and homogeneous model theory, which
however lack compactness.
We present some examples of these frameworks and discuss which properties
of the latter ones do or do not generalize to finitary AECs.
Nov. 20, 2007
Alf Dolich :
4 p.m. in SEO 427
Abstract
I will survey results on expansions of theories of
ordered structures by a generic unary predicate with a focus
on expansions of o-minimal theories.
Nov. 27, 2007
Christian Rosendal :
4 p.m. in SEO 427
Abstract
The geometric theory of Banach spaces underwent a tremendous
development in the decade 1990-2000 with the solution of several
outstanding conjectures by Gowers, Maurey, Odell and Schlumprecht.
Their discoveries both hinted at a previously unknown richness of the
class of separable Banach spaces and also laid the beginnings of a
classification program for separable Banach spaces due to Gowers.
However, since the initial steps done by Gowers, little progress was made
on the classification program. We shall discuss some recent advances due
to V. Ferenczi and myself on this by means of Ramsey theory and dichotomy
theorems for the structure of Banach spaces. This simultaneously allows us
to answer some related questions of Gowers concerning the quasiorder of
subspaces of a Banach space under the relation of isomorphic
embeddability.
Dec. 4, 2007
Moshe Kamensky :
4 p.m. in SEO 427
Abstract
The Galois theory of linear differential equations can be usefully
interpreted as an instance of the general model theoretic construction
of the automorphisms group of one definable set over another. I will
present an elementary version of this construction, which is useful for
interpreting the Galois group of a linear difference equation.
Jan. 22, 2008
John Goodrick :
1 p.m. in SEO 712
Abstract
Dp-minimal theories are a subclass of dependent theories that generalize weakly minimal theories in the stable context, arising from Shelah's "dp-ranks." Dp-minimality can be defined quite simply, and in the context of ordered structures it generalizes weak o-minimality. In a divisible abelian ordered group, dp-minimality is not the same as weak o-minimality, but we show that a slight weakening of the Monotonicity Theorem holds: any definable unary function is a union of finitely many continuous, locally monotonic (partial) functions.
Feb. 4, 2008
Craig Smorynski :
4 p.m. in SEO 427
Abstract
The genuine accomplishments of the Darmstadt school of nonstandard analysis has largely been overshadowed by the later developments of Robinson and his followers. The workings of the Darmstadters deserve more recognition and celebration.
Feb. 11, 2008
Itai Ben Yaacov :
4 p.m. in SEO 427
Abstract
A relation between two sets X and Y is said to be dependent if it does
not shatter arbitrarily large finite subsets of X. Similarly, a
function X x Y -> [0,1] is dependent if for every e > 0 it does not
e-shatter arbitrarily large finite subsets of X. A characterisation of
dependent functions in terms of the growth rate of the mean width of a
family of associated convex compacts is essential to proving that the
expectation of a random family of uniformly dependent functions is again
dependent. Model theoretically, this means that the Keisler randomisation of a
dependent theory is again dependent, answering a question of Pillay and
allowing a more direct approach to the study of Keisler measure in
(classical) dependent theories.
Feb. 18, 2008
Sergei Starchenko :
4 p.m. in SEO 427
Abstract
In this talk we present a model theoretic proof of the following
theorem due to Bieri and Groves.
Theorem.
Let $V\subseteq (\mathbb{C}^*)^n$ be a variety.
For $h>0$ let $A_h$ be the set $h\cdot log(|V|)$ and
$A_0= lim_{h->0} A_h$.
Then $A_0$ is a semi-linear cone in $\mathbb{R}^n$.
Feb. 25, 2008
Martin Koerwien :
4 p.m. in SEO 427
Abstract
This is the first of two talks about the descriptive set theoretic
notion of Borel Reducibility and in particular
its applications to model theory. We begin by a brief exposition
of the links to one of the outstanding conjectures in logic,
Vaught's Conjecture, and then give an overview over
some basic results, and some of the results presented in the paper
"A Borel Reducibility Theory for Classes of Countable Structures"
(H. Friedman and L. Stanley, JSL 54(3), 1989). Later on
(probably in the second talk), we will give an introduction to
so-called essentially countable equivalence relations.
March 3, 2008
Martin Koerwien :
4 p.m. in SEO 427
March 10, 2008
Daniel Groves :
4 p.m. in SEO 427
Abstract
I will discuss some aspects of Sela's work on the elementary theory of
free groups. I will probably particularly focus on the structure of
two-quantifier sentences, and the procedure for verifying them, as outlined
in Sela's second and fourth papers.
March 17, 2008
Alice Medvedev :
4 p.m. in SEO 427
Abstract
Consider the following rational dynamical system: $V$ is an affine
space, and $F: V \rightarrow V$ acts polynomially coordinate-wise:
$F(x, y, \ldots, z) = f(x), g(y), \ldots, h(z))$ for polynomials
$f, g, \ldots, h$. What subvatieties of $V$ are invariant under this
action?
In a model of ACFA, consider a union $S$ of definable minimal sets
$S_i := \{ x: \sigma(x) = f_i(x) \}$ for some polynomials $f_i$. What
is the pregeometry on $S$ given by the model-theoretic algebraic
closure operator?
This is the same question, and I have an answer.
April 7, 2008
Uri Andrews :
4 p.m. in SEO 427
April 14, 2008
David Lippel :
4 p.m. in SEO 427
Abstract
A "positive elimination theorem" is a statement that certain positive
existential formulas are equivalent to positive quantifier-free
formulas. Here is a classical example. Let X be a Zariski-closed subset
of a complex projective space. Concretely, X is the solution set of a
system of homogeneous polynomial equations; thus, X has a positive
quantifier-free definition in the language of rings. Let Y be a
coordinate projection of X, so Y has a positive existential definition.
Classical elimination theory says that Y is Zariski-closed, i.e. Y is
actually defined by a positive quantifier-free formula.
Prestel has proved some positive elimination results for valued fields,
working in a one-sorted language. I will discuss some generalizations to
two-sorted languages; these can be used to re-prove some basic facts in
tropical geometry. This is joint work with Matthias Aschenbrenner and
Sergei Starchenko.
April 21, 2008
Alf Dolich :
4 p.m. in SEO 427
April 28, 2008
Krzysztof Krupinski :
4 p.m. in SEO 427
Abstract
I will talk about some results yielding infinite interpretable fields in rosy groups of finite thorn U-rank. These results generalize some theorems from the finite Morley rank case and from o-minimal structures. I will prove the existence of such fields in the presence of certain V-definable rings (generalizing a result by Peterzil and Starchenko for o-minimal structures) and in a situation when an infinite, definable abelian group acts definably as a group of automorphisms on a definable abelian group. The interesting fact is that the lack of most of the tools (such as the uniform chain condition on intersections of uniformly definable subgroups or Zilber's Indecomposables theorem) has forced me to use completely fundamental tools (such as the compactness theorem and basic properties of dimension), and as a result I have obtained simpler proofs than those in the finite Morley rank case or o-minimal structures. Using these results, I have proved the existence of an infinite interpretable field in any solvable-by-finite but not nilpotent-by-finite group of finite thorn U-rank satisfying NIP.
Aug. 26, 2008
Inessa Epstein :
4 p.m. in SEO 612
Abstract
We consider a countable group G acting in a Borel way by measure preserving automorphisms on a standard probability space X. The orbits of this action give rise to an equivalence relation on X. We say two measure preserving actions of groups G and H on spaces X and Y, respectively, are orbit equivalent if there is a measure preserving bijection between conull subsets of X and Y identifying the orbits. We discuss the complexity of the classification problem of free, measure preserving, ergodic actions of a countable group under orbit equivalence and show that it is not classifiable by countable structures.
Sept. 2, 2008
Ahuva Shkop :
4 p.m. in SEO 612
Sept. 16, 2008
Paul Larson :
4 p.m. in SEO 612
Abstract
Speakers who do not submit abstracts speak about Three Blind Alligators.
Sept. 25, 2008
John Baldwin :
4 p.m. in SEO 712
Sept. 30, 2008
Rahim Moosa :
4 p.m. in SEO 612
Abstract
A compact complex manifold $M$ is viewed as a model-theoretic structure in the language where there is a predicate for each analytic subset of $M^n$. The manifold is essentially saturated if it admits a countable sub-language from which all complex-analytic subsets are definable (with parameters). All compact Kaehler manifolds (and their holomorphic images, the Kaehler-type spaces) are essentially saturated. I will describe some recent joint work with Ruxandra Moraru and Matei Toma in which we show that the converse is not true. We show that Inoue surfaces of type $S_M$ are essentially saturated (though not of Kaehler-type).
Oct. 7, 2008
Clifton Ealy :
3 p.m. in SEO 612
Abstract
It has been conjectured that superrosy dependent fields should be either algebraically closed or real closed. I will discuss recent work on this problem, with a focus on possible counterexamples.
Oct. 14, 2008
Alexey Ovchinnikov :
3:30 p.m. in SEO 612
Abstract
Perfect difference ideals are algebraic objects that
correspond to solutions of systems of algebraic difference
equations generalizing the algebraic-geometric correspondence
between solutions and radical ideals. Difference elimination
allows to reveal important information about the ideal and
the system, for example, its consistency. Currently, there
are some algorithms that do so only for ordinary difference
equations. We will be discussing those and will look at
the structure of perfect difference ideals.
Oct. 21, 2008
Bektur Baizhanov :
4 p.m. in SEO 612
Abstract
In my talk I present our (with Viktor Verbovskiy) generalization
of the notion o-minimality, which allows us to apply technics of
stability theory to investigation of ordered structures.
Let $T$ be a complete theory, having $\emptyset$-definable
relation of linear order. Let $M$ be a model of $T$, $M$ be a
model of $T$, $A\subseteq M$. For any 1-formula $\phi(x)$ we
define a convex hull $C_{\phi} (x)$ as $ C_{\phi}(x) \stackrel{\triangle}{=}
\exists y,z (\phi(y) \land \phi (z) \land y \le x \le z) $; and
for an one-type $p\in S^1(A)$ we define a convex hull $c(p)$ as $
c(p) \stackrel{\triangle}{=} \{ C_{\phi} \mid \phi \in p\}. $
Denote $S^1_{c(p)}(A)=\{q\in S^1(A)| c(p)\subseteq q\}$. The
model $M$ is o-stable in $\lambda$ if for all $A\subseteq M$,
$|A| \le \lambda$, for any 1-type $p$ $|S^1_{c(p)}|\le \lambda.$
Theory $T$ is o-stable, if every model of $T$ is. As in stability
theory we can introduce the notions of o-$\omega$-stability and
o-superstability. Notice that Morley's Theorem for
o-$\omega$-stability holds. It follows from definition that
o-minimal theory and weakly o-minimal theory are o-$\omega$-stable
and quasi o-minimal theory is o-superstable.
<b>Theorem 1</b> Let $T$ be o-stable theory, then $T$ does not
have the independence property.
<b>Theorem 2</b> Any infinite model $(M;=, <)$ has
o-superstable theory.
B. Baizhanov and V. Verbovskiy <i> O-stable theories</i>, preprint, 2008.
Oct. 28, 2008
Julia Knight :
4 p.m. in SEO 612
Abstract
The real closed ordered fields are the models of the theory of the ordered field of reals. An integer part sits in $R$ in the way that the integers sit in the reals. More precisely, an integer part for an ordered field $R$ is a discrete ordered ring $I\subseteq R$ such that $1$ is the first positive element, and for each $x\in R$, there exists $i\in I$ such that $i\leq x < i+1$. Shepherdson showed that $I$ is an integer part for a real closed ordered field iff it is a model of $IOpen$---the fragment of arithmetic with induction axioms just for open formulas. It is natural to ask when a real closed ordered field $R$ has an integer part satisfying full $PA$. Paola D'Aquino, Sergei Starchenko, and I showed that if $R$ has an integer part $I$ which is a nonstandard model of $PA$, then $R$ is recursively saturated, with types determined by $I$. Hence, if $R$ is countable, then $I$ determines the isomorphism type. We also showed that if $R$ is recursively saturated, then there is an integer part $I$ satisfying $PA$ such that $R$ is the real closure of $I$.
Mourgues and Ressayre, with some help from Marker and Delon, showed that every real closed ordered field has an integer part. Currently, Karen Lange and I are considering the complexity of integer parts.
Oct. 30, 2008
Daniel Mauldin :
4:30 p.m. in SEO 636
Abstract
I plan to discuss some of the results and unsolved problems
relating to: Steinhaus' lattice problem or simultaneous tiling problem: Is
there a set in the plane which meets each isometric copy of Z^2 in exactly
one point? and Sierpinski's 2 point problem: Is there a set in the plane
which meet each straight line in exactly two points?
Nov. 4, 2008
Christian Rosendal :
4 p.m. in SEO 612
Abstract
I will present the main ideas of a new proof of the fundamental block
Ramsey theorem of Gowers that has had spectacular applications in Banach
space theory. Gowers' original proof cleverly combined Ramsey theory and
game theory to compensate for the fact that a true Ramsey theoretical
statement fails to hold in general, but at the same time involved
approximation arguments compensating for a lack of compactness. This made
the proof somewhat complicated and the ensuing notion of weakly Ramsey
sets involved quantifications over approximations hard to induct over and
extend beyond analytic sets. Our new proof involves a new game, the
infinite asymptotic game, and a new notion of strategically Ramsey sets
which completely echews approximations and allows us to give a very smooth
proof of Gowers' theorem.
Nov. 11, 2008
Slawomir Solecki :
4 p.m. in SEO 612
Abstract
I will present an extension of Proemel's dual structural Ramsey theorem to structures with functions as well as relations. I will explain how this theorem (and its possible further extensions) relate to dynamics of the homeomorphism group of the pseudo-arc. This explanation will involve a dualization of Fraisse limits that I developed jointly with Trevor Irwin.
Nov. 18, 2008
Serge Randriambololona :
4 p.m. in SEO 612
Abstract
To each o-minimal expansion of a (real closed) field, one can associate the set
of germs at infinity of its unary functions, which form a Hardy field.
Valuational properties of these Hardy fields give good information about the
initial structure.
Motivated by a conjecture of van den Dries and a result of F.-V. and S.
Kuhlmann, I will discuss
whether an o-minimal expansions of the field of the reals is, in general, fully
determined by its associated Hardy field.
Dec. 2, 2008
Lou van den Dries :
4 p.m. in SEO 612
Abstract
Most of what I will talk about is work by Vinicius C.L.
and will be part of his thesis. Vector spaces over division rings
are in general not pseudo-finite, but share striking properties with pseudo-finite structures. This can be proved by means of a
canonical finitely additive measure on the category of
definable sets in such a space. This measure takes polynomial values.
Much of this extends to the eq-expansion of a vector space, and some of
it to similar structures, like strongly minimal groups.
Dec. 10, 2008
Jeff Burdges :
1 p.m. in SEO 612
Abstract
We will discuss the generation principle which says that the connected centralizers of the involutions in a four-group generate the whole connected group. The generation principle is one of the core structural properties in the generic case and much of the quasi-thin case, meaning these identifications results make significant use while the proof requires the full uniqueness case analysis. We give a shorter proof in the case that G has no unipotent torsion.
Jan. 13, 2009
Asger Tornquist :
4 p.m. in SEO 612
Abstract
In von Neumann algebra theory, a factor is a von Neumann algebra in which the center consists of multiples of the identity. Factors make up the building blocks out of which any other von Neumann algebra can be build. The problem of classifying von Neumann factors is therefore as old as the subject itself.
In this talk I will discuss a recent result (joint with Roman Sasyk, Buenos Aires), where we show that separable von Neumann factors are not classi able by a reasonable assignment of invariants that are countable structures, in particular, there is no suitably "Borel" assignment of countable groups, graphs or other countable structures as complete invariants for the isomorphism relation of separable factors. The proof involves among other things Greg Hjorth's theory of turbulence, the group-measure space construction, and the deformation/rigidity techniques developed by Sorin Popa.
Jan. 14, 2009
Maryanthe Malliaris :
4 p.m. in SEO 612
Abstract
A natural and deep move in model theory is to classify theories into complexity classes according to various, often quite coarse, criteria. If all goes well these criteria act as a kind of scaffolding: they call attention to deep structural properties of the underlying models or theories which, it turns out, can already be seen locally within the theory. Shelah's stability theory (counting types over sets) is a beautiful and well-known example. Keisler's order might be another, but parts of its structure have remained elusive.
The talk will have two complimentary parts: on one hand, I'll describe recent progress on Keisler's order and the structure theory it suggests, and on the other, a project to build from within first-order theories a framework in which these basic combinatorial issues are visible and can be analyzed.
Jan. 20, 2009
Uri Andrews :
4 p.m. in SEO 612
Abstract
A model of a strongly minimal theory can be characterized by
its dimension. The spectrum of a strongly minimal theory is the set of
dimensions of models that have recursive presentations. There is a
standing open question of which subsets of $\omega+1$ can be achieved
as a spectrum of a strongly minimal theory. In this talk, we will
examine some of the spectra already known possible and will explore
whether they can be attained in a theory with a finite language.
Jan. 27, 2009
Konstantin Slutsky :
4 p.m. in SEO 612
Abstract
Several years ago Kechris and Rosendal proved the existence of the comeagre conjugacy class in the group of the homeomorphisms of the Cantor set. Their proof was rather abstract. We'll discuss the explicit construction of the homeomorphism with comeagre conjugacy class due to Akin, Glasner and Weiss.
Feb. 3, 2009
John Baldwin :
4 p.m. in SEO 612
Abstract
We discuss the general question. If $A$ is a subset of $M$, does naming $A$ change the stability class?
We consider sufficient conditions provided (in various combinations) by Baizhanov, Baldwin,
Benedikt,Bouscaren, Casanovas, Poizat, Shelah, Ziegler for the answer to be NO.
And we consider specifi c conjectures for extending these results.
E.g. Conjecture: If $M$ is stable and
$I$ is a set indiscernibles in $M$, then $(M; I)$ is stable.
Baizhanov-Baldwin have proved yes if $I$ has infi nite co-dimension.
Feb. 10, 2009
Christian Rosendal :
4 p.m. in SEO 612
Abstract
An old problem due to Christensen asks whether any universally
measurable homomorphism (i.e., measurable with respect to any Borel
probability measure) between Polish (i.e. separable, complete metric)
groups is continuous. This was originally settled for the group of real
numbers by Steinhaus and later for any second countable locally compact
group by Weil in the first half of the 20th century. However, since
locally compact groups are exactly those that admit translation invariant
measures, the situation for arbitrary Polish groups is radically
different. Nevertheless, Steinhaus and Weil's result was extended to
Abelian groups by Christensen in the late 1960s via his notion of Haar
null sets in arbitrary Polish groups and subsequent work mainly by Solecki
has further extended this to larger classes of Polish groups. I will give
an introduction to the basic theory and also present some new results
pointing towards a positive answer to Christensen's problem.
Feb. 17, 2009
Christian Rosendal :
4 p.m. in SEO 612
Abstract
An old problem due to Christensen asks whether any universally
measurable homomorphism (i.e., measurable with respect to any Borel
probability measure) between Polish (i.e. separable, complete metric)
groups is continuous. This was originally settled for the group of real
numbers by Steinhaus and later for any second countable locally compact
group by Weil in the first half of the 20th century. However, since
locally compact groups are exactly those that admit translation invariant
measures, the situation for arbitrary Polish groups is radically
different. Nevertheless, Steinhaus and Weil's result was extended to
Abelian groups by Christensen in the late 1960s via his notion of Haar
null sets in arbitrary Polish groups and subsequent work mainly by Solecki
has further extended this to larger classes of Polish groups. I will give
an introduction to the basic theory and also present some new results
pointing towards a positive answer to Christensen's problem.
Feb. 24, 2009
C. Ward Henson :
4 p.m. in SEO 612
Abstract
In general, this talk is about separable metric structures whose theories in continuous logic are $\omega$-categorical and admit quantifier-elimination. There are several interesting examples that arose "in nature." There are good characterizations of $\omega$-categoricity (when the signature is countable) and of QE. The Fraisse construction has a natural extension to the metric setting and it has produced a few more examples. Moreover, there are lots of interesting open questions. In particular, the classification program that has generated so much interesting mathematics in the classical setting has not yet been taken up in a serious way.
March 3, 2009
Alexander Levin :
4 p.m. in SEO 612
Abstract
In this talk we consider the problem of compatibility of difference field extensions, that is, the problem of K-embedding of two difference field extensions L/K and M/K into some difference overfield of K. We are going to start with a brief discussion of the compatibility of classical and differential field extensions. Then we will consider the situation in difference algebra where even two extensions of an ordinary difference field of zero characteristic can be incompatible. After introducing the concepts of limit degree and core of a difference field extension and describing their properties, we will prove a criterion of compatibility for extensions of ordinary difference fields.
March 10, 2009
Sara Quinn :
4 p.m. in SEO 612
Abstract
In this talk I will give evidence that the standard back-and-forth relations are a powerful tool in computable structure theory. I will give all necessary background and definitions, and then give two results on equivalence structures that can be proved using the back-and-forth relations. These two results are from my dissertation, and involve index set complexity, Scott sentences, and Turing computable embedding.
March 17, 2009
David McClendon :
4 p.m. in SEO 612
Abstract
A theorem of Becker and Kechris guarantees the existence of "nice
topologies" for jointly Borel Polish group actions on Polish spaces (a
"nice topology" is a Polish topology on the phase space with the same
Borel sets as the original topology for which the action is jointly
continuous). The same result holds for countably generated actions of
Polish semigroups.
But for semiflows (actions of the semigroup $[0,\infty)$ of non-negative
real numbers), this theorem is false. Let $\{T_t : t \geq 0\}$ be a
semiflow and consider two points $x \neq y$ in the phase space which map
to the same point under all $T_t, t > 0$. We say $x$ and $y$ are
"instantaneously and discontinuously identified" by the semiflow; the
presence of any such pair of points ensures that no "nice topology" can
exist for the semiflow. In this talk, we will discuss some results
related to this phenomenon and explain why this behavior is worth
studying, from the perspective of ergodic theory.
March 30, 2009
Salma Kuhlmann :
4 p.m. in SEO 612
Abstract
We consider a totally ordered set $\Gamma$ of cardinality $\aleph_1$, of which elements are germs at $+\infty$ of real valued functions of a real variable. We show that the order type of $\Gamma$ is that of a lexicographic ordering which admits $2^{\aleph_1}$ automorphisms of pairwise distinct orbital growth. We associate to each such automorphism a well defined logarithmic function on the field $\mathbb{R}((G))_{\aleph_1}$, where $\mathbb{R}((G))_{\aleph_1}$ is the field of generalized series with countable support, real coefficients and exponents in the group $G$ of transmonomials at $+\infty$ defined by $\Gamma$. We show that distinct automorphisms induce logarithmic functions of distinct growth rates.
April 7, 2009
Dave Sahota :
4 p.m. in SEO 612
Abstract
Friedman and Stanley showed that fields are Borel complete. We
will survey key examples leading to this result.
April 14, 2009
Alexei Kolesnikov :
4 p.m. in SEO 612
Abstract
A recent work of Hrushovski links failure of certain amalgamation properties to definability of groupoids in stable theories. In a joint work with John Goodrick, we provide an explicit construction of such groupoids, show that the groupoids can be non-trivial even in a totally categorical theory, and obtain a way to ``eliminate'', in a certain sense, such groupoids by adding additional sorts to models of the theory. I will describe the results and outline our current research problems.
April 21, 2009
Chris Laskowski :
4 p.m. in SEO 612
Abstract
We begin by discussing notions of compressions
occurring in computational learning theory and relate these
to the definability of types. We offer an improvement to
Shelah's theorem for stable formulas and investigate the conjecture
that dependent formulas have uniform type definitions over finite
sets.
Parts of this are joint with Hunter Johnson and Vince Guingona.
April 28, 2009
Ben Miller :
4 p.m. in SEO 612
Abstract
Since its inception, the study of definable subsets of the real numbers has been dominated by a variety of structural dichotomy theorems. In recent times, the proofs of these theorems have grow increasingly complex and dependent upon techniques from mathematical logic. After giving a brief history of the subject, I will discuss a new approach to giving classical proofs of these results which is motivated by ideas from graph theory.
May 5, 2009
Charles Steinhorn :
4 p.m. in SEO 612
Abstract
It has been widely thought that o-minimal structures might lie at the first level of a hierarchy of ordered structures, in analogy with strongly minimal structures. In joint work with A. Onshuus, we develop a framework for ordered structures of finite rank. In particular, we analyze linear orders definable in o-minimal structures, and this will be the focus of the first part of the talk. This analysis appears to have an interesting application in mathematical economics---joint work with T. Brihaye, C. Michaux, and Onshuus---discussion of which is the subject of the second part of the talk.
May 18, 2009
John Baldwin :
10:30 a.m. in SEO 612
Abstract
Hrushovski generalized the Fraisse construction to provide counter examples to two major conjectures in model theory. Baldwin and Shelah adapted one of these arguments to give the first full proof of the Spencer-Shelah 0-1 laws for random graphs with edge probability \$n^{-\alpha}\$. Recently, Baudisch, Hils, Martin-Pizzaro and Wagner used the method to construct expansions of algebraically closed fields with a definable subgroup of the multiplicative group. We will try to give an organized account of the 40 or 50 constructions using this method and suggest some further open problems. In particular, we will describe the essential ideas of Laskowski's significantly simplification of the random graph argument.
July 12, 2009
John Baldwin :
10 a.m. in SEO 636
Abstract
This an introductory talk for a miniconference on the role of set theory in model theory.
James Freitag :
11:30 a.m. in SEO 636
Abstract
Proof of weak diamond from weak gch
Fred Drueck :
2 p.m. in SEO 636
Abstract
If an AEC is categorical in $\lambda$ and has few models in $\lambda^+$ then it has amalgamation in $\lambda$.
July 13, 2009
Monica VanDieren :
4:15 p.m. in SEO 636
Abstract
Recent work of Grossberg, VanDieren, and Villaveces
Paul Larson :
10 a.m. in 305 Taft Hall
Abstract
Appendix to Second Edition of Proper Forcing
Martin Koerwien :
11:30 a.m. in 305 Taft Hall
Abstract
Discussion of putative counterexamples.
Alf Dolich :
2 p.m. in SEO 636
Abstract
A discussion of the techniques of set theoretic absoluteness and their role in model theory.
Aug. 25, 2009
Alice Medvedev :
4 p.m. in SEO 612
Abstract
Instead of thinking of a model of ACFA as a field with one automophism $\sigma$, one can think of it as an action of $(\mathbb{Z}, +)$ on the field, where $n$ acts by $\sigma^n$. I will say a few words about what happens when $\mathbb{Z}$ is replaced with $\mathbb{Q}$.
Sept. 1, 2009
Maryanthe Malliaris :
4 p.m. in SEO 612
Abstract
The characteristic sequence of hypergraphs $\langle P_n : n<\omega
\rangle$ associated to a formula $\phi(x;y)$, defined by $P_n(y_1,\ldots
y_n) = (\exists x) \bigwedge_{i\leq n} \phi(x;y_i)$, is a tool for
studying the combinatorial complexity of $\phi$-types. This talk will
discuss how graph-theoretic techniques, notably Szemeredi regularity,
can be naturally applied to the study of model-theoretic complexity via
the characteristic sequence.
Sept. 8, 2009
Ahuva Shkop :
4 p.m. in SEO 612
Abstract
In the 50's, Shapiro conjectured that if two exponential
polynomials in one variable which are each sums of terms of the form
exp(a+bz) have no common factors, then they have only finitely many common
zeros. The goal of this talk is to prove that a special case of this
conjecture holds in Pseudoexponentiation as well as in any other
algebraically closed exponential field of characteristic zero satisfying
Schanuel's conjecture.
Sept. 15, 2009
Salma Kuhlmann :
4:30 p.m. in SEO 612
Abstract
We consider the valued field ${K}:=\mathbb{R}((\Gamma))$ of generalised series (with real coefficients and monomials in a totally ordered multiplicative group $\Gamma\>$). We investigate how to endow ${K}$ with a series derivation, that is a derivation that satisfies some natural properties such as commuting with infinite sums (strong linearity) and (an infinite version of) Leibniz rule. We characterize when such a derivation is of Hardy type, that is, when it behaves like differentiation of germs of real valued functions in a Hardy field. We provide a necessary and sufficent condition for a series derivation of Hardy type to be surjective.
(This is joint work with Mickael Matusinski.)
Sept. 22, 2009
Matthias Aschenbrenner :
4 p.m. in SEO 612
Abstract
A classical result due to Kirszbraun (1934), which plays an important role in geometric measure theory, shows that every Lipschitz map $S\to\mathbb R^n$ on a subset $S$ of $\mathbb R^m$ can be extended to a Lipschitz map $\mathbb R^m\to\mathbb R^n$ with the same Lipschitz constant. The usual proofs of this theorem in the literature employ, in some form or other, the Axiom of Choice. We discuss a definable version of this result. (Joint with Andreas Fischer.)
Sept. 29, 2009
Fifth midwest computability seminar :
4 p.m. in SEO 612
Oct. 6, 2009
Moshe Kamensky :
4 p.m. in SEO 612
Abstract
I will explain how the model theoretic notions of internality and the binding group can be applied to (and viewed as a generalisation of) the Tannakian formalism, i.e., the description of the category of representations of an affine algebraic group.
Oct. 13, 2009
Antonio Montalban :
4 p.m. in SEO 612
Abstract
The proof theoretic strength of the various levels of Determinacy
have been studied for the last 40 years in computability theory, set
theory and reverse mathematics. With Richard Shore, we have recently
found the exact level at which determinacy becomes non-provable in
Second Order Arithmetic.
In this talk we will discuss the history of the subject and the new results.
Oct. 20, 2009
Joseph Flenner :
4 p.m. in SEO 612
Abstract
While logic has produced many results about the $p$-adics, among them a decision procedure due to Paul Cohen, the general theory of henselian valued fields presents an inherent difficulty: they are built on structures of arbitrary complexity in the residue field and value group. Ax-Kochen and Ersov, however, proved their completeness result for some henselian valued fields relative to the theories of the residue field and value group, and more recently, there have been some relative quantifier elimination theorems of Kuhlmann. In this spirit, we describe a structure of \emph{leading terms} associated to a valued field, and outline a proof of decidability for henselian valued fields of characteristic $0$ relative to the leading term structures.
Oct. 27, 2009
Karen Lange :
4 p.m. in SEO 612
Abstract
An integer part $I$ of a real closed field $R$ is a discrete ordered subring containing $1$ such that for all $r\in R$ there exists a unique $i\in I$ with $i\leq r < i+1$. Mourgues and Ressayre [1] showed that every real closed field $R$ has an integer part. Let $k$ be the residue field of $R$, and let $G$ be the value group of $R$. Let $k\langle\langle G\rangle\rangle$ be the set of generalized power series of the form $\Sigma_{g\in S}a_gg$ where $a_g\in k$ and the support of the power series $S\subseteq G$ is well ordered. Mourgues and Ressayre produce an integer part of $R$ by building an isomorphism between $R$ and a truncation closed subfield of $k\langle\langle G\rangle\rangle$. We refer to the image of $r\in R$ as its development. In order to understand the complexity of integer parts, we analyze an algorithmic version of the Mourgues and Ressayre construction.
We consider the case where $R$ is countable, and we consider a list of the elements of a transcendence base for $R$ over $k$. Given such a list $\{r_1, r_2,\ldots \}$, the Mourgues and Ressayre construction becomes canonical. Let $R_n$ be the real closure of $k(r_1,\ldots,r_n)$. By a result of Shepherdson [2], the elements of $R_1$ have developments of length at most $\omega$. We show that elements of $R_n$ have developments of length at most $\omega^{\omega^{(n-1)}}$. Thus, the elements of $R$ have developments of length less than $\omega^{\omega^\omega}$. These bounds are sharp. This is joint work with Julia Knight.
References:
[1] M. H. Mourgues and J.-P. Ressayre, "Every real closed field has an integer part," J. Symb. Logic, vol. 58 (1993), pp. 641-647.
[2] J. Shepherdson, "A non-standard model for the free variable fragment of number theory", Bulletin de l'Academie Polonaise Des Sciences, vol. 12(1964), pp. 79-86.
Nov. 3, 2009
John Goodrick :
4 p.m. in SEO 612
Abstract
Recently (in joint work with Chris Laskowski) we characterized countable, weakly minimal theories $T$ such that any two (elementarily) bi-embeddable models of $T$ are isomorphic. We prove that
if $T$ is countable and weakly minimal, the following are equivalent:
1. $T$ has two bi-embeddable but nonisomorphic models;
2. There is an automorphism $f$ of the monster model of $T$ and a strong type $p$ over the empty set which is almost-orthogonal to $f(p) \otimes \ldots \otimes f^n(p) $ for any n.
3. $T$ has an infinite collection of models that are pairwise bi-embeddable but pairwise nonisomorphic.
The proof involves some geometric stability theory plus a Dushnik-Miller type argument to build nonisomorphic models by "killing'' every potential isomorphism at each stage of the construction.
Nov. 10, 2009
Justin Moore :
4 p.m. in SEO 612
Abstract
While it is not known whether Thompson's group $F$ is amenable, I will
establish a lower bound on the cardinality of its Foelner sets. In
particular, I will demonstrate the following: There is a constant $C > 1$
such that if $A$ is a $C^{-n}$-Foelner set in $F$, then $A$ contains at
least $H(n)$ elements, where $H(0)=0$ and $H(n+1)=2^{H(n)}$.
Nov. 12, 2009
Uri Andrews :
4:15 p.m. in SEO 612
Abstract
There is a theory which computes arithmetic, is strongly minimal, and all of its models have recursive presentations.
We will preview the Hrushovski construction and some ways to alter the construction to code non-recursive information. Then we will use an infinite worker argument to show that with these method we can code the most complicated set possibly coded in a recursive structure, namely the set of true statements in Arithmetic ($0^{\omega}$).
Nov. 17, 2009
Lynn Scow :
4 p.m. in SEO 612
Abstract
In the 1970s S. Shelah gave the following characterization of stable theories: a theory is stable if and only if any indiscernible sequence in a model of the theory is an indiscernible set. I will present a similar characterization of NIP theories, as theories in which any random ordered graph-indiscernible in a model of the theory remains indiscernible strictly with respect to the order. In this talk I will explain what I mean by a random ordered graph-indiscernible and I will indicate how the result is proved using the Nesetril-Rodl theorem. If time permits, I will discuss an additional example of a characterization of stable theories by generalized indiscernibles that generalizes more faithfully on Shelah's.
Dec. 1, 2009
Meghan Anderson :
4 p.m. in SEO 612
Abstract
The theory of valued D-fields provides an interesting setting for the study of difference and differential Galois groups. The extra structure provided by the valuation can be used to relate groups arising from these two types of equations; however, these groups are not always what one might expect. I will talk about the some of the advantages and limitations of working in this particular theory, and look at a few specific equations.
Jan. 12, 2010
No seminar this week. :
4 p.m. in SEO 612
Jan. 19, 2010
Roman Kossak :
2:30 p.m. in SEO 612
Abstract
I will briefly review the basics of Borel reducibility theory and I will discuss its applications to model theory of Peano Arithmetic. I will focus on the isomorphism problem for finitely generated models and on the conjugacy problem for various expansions of countable recursively saturated models.
It is joint work with Samuel Coskey.
Jan. 26, 2010
Alexey Ovchinnikov :
4 p.m. in SEO 612
Abstract
In this talk, we apply Dima Trushin's work on difference Nullstellensatz for difference closed pseudofields to construct a Galois theory of difference equations with difference parameters. Existing Galois theories for difference and differential equations with differential parameters developed by Cassidy, Singer, and Hardouin are based on differentially closed fields. However, in our case difference closed fields are not a valid substitute due to their lack of quantifier elimination, and, therefore, we are using difference simple rings that are products of fields (called pseudofields) instead.
Feb. 2, 2010
John Baldwin :
4 p.m. in SEO 612
Abstract
We provide some context for a number of variants on the Ehrenfeucht-Mostowski construction and expound the construction of tree indiscernibles for sentences of $L_{\omega_1,\omega}$. Two results of Baldwin-Shelah on the stability spectrum for $L_{\omega_1,\omega}$ use these methods.
Here ${S_i}^m(M)$ denotes an appropriate notion ($at$ or ${mod}$) of Stone space of $m$-types over $M$.
Theorem A. Suppose that for some positive integer $m$ and for every $\alpha< \delta(T)$,
there is an $M \in K$ with $|{S^m}_i(M)| > |M|^{beth_\alpha(|T|)}$.
Then for every $\lambda \geq |T|$, there is an $M$ with $|{S^m}_i(M)| > |M|$.
Theorem B. Suppose that for every $\alpha<\delta(T)$, there is $M_\alpha \in K$ such
that $\lambda_\alpha = |M_{\alpha}| \geq beth_\alpha$ and $|{S^m}_i(M_\alpha)| >
\lambda_\alpha$. Then for any $\mu$ with $\mu^{\aleph_0}>\mu$, $K$ is not $i$-stable in $\mu$.
These results provide a new kind of sufficient condition for the unstable case and shed some light on the spectrum of strictly stable theories in this context. The methods avoid the use of compactness in the theory under study.
Feb. 9, 2010
Christian Rosendal :
4 p.m. in SEO 612
Abstract
We give an introduction to the interactions between strong versions of residual finiteness for groups and finite approximability of actions. As an application, we shall show that the isometry group of the rational Urysohn metric space has ample commuting generics.
Feb. 16, 2010
Clifton Ealy, Jana Marikova, Chris Miller :
11 a.m. in SEO 636
Abstract
Clifton Ealy: Thorn-Forking in Continuous Logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example of an essentially continuous unstable theory with a nice notion of independence. We also obtain a new fact for classical real rosy theories: a real rosy theory which has weak elimination of finitary imaginaries is rosy with respect to finitary imaginaries. (Joint work with Isaac Goldbring.)
Jana Marikova: Valuations on o-minimal fields. Let R be an o-minimal field and V a proper convex subring. We show that the o-minimality of the corresponding residue field with structure induced from R via the residue map is equivalent to (R,V) satisfying a first-order axiom scheme.
Chris Miller: A continuous extension property for o-minimal expansions of ordered groups. Let $f:R^n \rightarrow R$ be bounded and definable in an o-minimal expansion of an ordered group R. Then there are finitely many cells $C_i$ in $R^n$ whose union has interior and the frontier of each $C_i$ contains the origin and $f$ extends continuously to the closure of $C_i$. This gives an open set U of real numbers such that the expansion of the real exponential field by U Borel-interprets the real projective hierarchy, yet every definable set either has interior or is nowhere dense.
Feb. 22, 2010
Martin Hils :
4 p.m. in SEO 612
Abstract
The greed field of Poizat is an algebraically closed field $K_\omega$ together with a proper infinite divisible torsion free subgroups $U$ of the multiplicative group. It is of Morley rank $\omega \cdot 2$ and obtained by Hrushovski's amalgamation method. It may be collapsed into a ``bad field'', i.e. a finite Morley rank analogue $(K_\mu, 0, 1, +, \times, U)$. Recall that if T is a model-complete $L$-theory, the generic automorphism is said to be axiomatizable in T is the class of existentially closed models of the theory $T_\sigma = T + (\sigma \mbox{ is an automorphism})$ is elementary. In the talk we show that the generic automorphism is axiomatisable in the theory of green fields of Poizat (once this theory is Morleyised) as well as in the theory of the bad field. As a corollary, we obtain ``bad pseudofinite fields'' in characterisitc $0$. In both cases, we give geometric axioms. In fact, there is a general framework aloowing this kind of axiomatisation. There are serious definability issues related to the necessary ``choice of (unique) gree roots'', and Kummer thoery comes into play. We overcome these difficulties using weak CIT and an effective version of an argument by Zilber, showing that being Kummer-generic is a defiable property for algebraic varieties in characteristic $0$. Finally, we will discuss similar results in various other theories obtained by Hrushovski amalgamation -- both non-cllapsed and collapsed: the fusion of two strongly minimal thoeries, black fields in all characteristics, red fields in positive characteristic etc. The proofs are less involved since the above definability problems do not arise in these contexts.
Feb. 23, 2010
Ahuva Shkop :
11 a.m. in SEO 612
March 2, 2010
Prerna Juhlin :
4 p.m. in SEO 612
Abstract
Superstable theories of finite rank can be "built" using realizations of semiminimal types. Using a level-partitioning of semiminimal constructions, we study when dependence above the first level has a modular-like behavior--a property we formalize and call the Level Dependence Property (LDP). LDP is equivalent to the Canonical Base Property of Moosa and Pillay. It has been shown that the property holds in compact complex spaces, differentially closed fields, and difference fields. We prove that in superstable theories of finite rank, LDP holds under certain orthogonality and rank assumptions.
March 9, 2010
James Freitag :
4 p.m. in SEO 612
Abstract
We work over a differential field. For points on differential varieties, there is a topological notion of genericity and a model theoretic notion of genericity. We will talk about cases where these notions agree (groups, low transcendence degree) and an example where the collections of points satisfying the two different notions are actually disjoint.
March 16, 2010
Konstantin Slutsky :
4 p.m. in SEO 612
Abstract
One of the reasons for the interest in Fraisse classes is the richness of the groups of automorphisms of their limits. We shall discuss groups of automorphisms of the limits of certain linearly ordered Fraisse classes and show that all topological similarity classes in them are meager. Examples will include: groups of automorphism of the rationals (as a linear ordering), of the randomly ordered random graph, and of the randomly ordered Urysohn space. The talk will be based on a joint work with Christian Rosendal.
April 6, 2010
Sergei Starchenko :
4 p.m. in SEO 612
Abstract
In this talk I will discuss relations between definable compactness, fractional Helly number and VC-densities in NIP theories.
April 13, 2010
Salih Azgin, Amador Martin Pizarro :
1 p.m. in SEO 636
Abstract
see http://www.math.uic.edu/~alice/mwmtday.html
* 11:15am: Lunch at Joy Yee's (1335 S. Halsted)
* 1pm: One talk
* 2:30: Two talk
* 4pm: Three talk
* 5:45pm: Dinner at Greek Islands
April 20, 2010
Slawomir Solecki :
4 p.m. in SEO 612
Abstract
I will talk about the following result: for a Polish group G of isometries of a locally compact separable metric space, each measure preserving Boolean action by G has a spatial model or, in other words, has a point realization. This result extends both a classical theorem of Mackey and a recent theorem of Glasner and Weiss, and it covers interesting new examples. In order to prove this result, I will give a characterization of Polish groups of isometries of locally compact separable metric spaces which may be of independent interest. The solution to Hilbert's fifth problem plays an important role in establishing this characterization.
This is a joint work with A. Kwiatkowska.
April 27, 2010
Reed Solomon :
4 p.m. in SEO 612
Abstract
A computable structure A is called computably categorical if for every computable structure B which is isomorphic to A, there is a computable isomorphism between A and B. Similarly, a computable structure A is called relatively computable categorical if for every isomorphic copy B (not necessarily computable), there is an isomorphism between A and B computable in the degree of B. It is known that relatively computably categorical structures have particularly simple Scott families consisting of finitary formulas. We construct an example showing that this property can fail as badly as possible for computable categorical structures.
May 4, 2010
William Simmons :
4 p.m. in SEO 612
Abstract
Hrushovski's proof of the Mordell-Lang Conjecture for function fields uses hard results about Zariski geometries. Pillay and Ziegler developed a simpler method employing differential jet spaces that implies the result Hrushovski needs in characteristic zero. I describe Pillay and Ziegler's setting and sketch several proofs (e.g., of the Zilber dichotomy for differentially closed fields).
Aug. 24, 2010
Christian Rosendal :
4 p.m. in SEO 612
Aug. 31, 2010
Valentin Ferenczi :
4 p.m. in SEO 612
Sept. 7, 2010
Lynn Scow :
4 p.m. in SEO 612
Sept. 14, 2010
Isaac Goldbring :
4 p.m. in SEO 612
Abstract
Urysohn's metric space U is the unique (up to isometry) Polish (i.e. complete, separable) metric space which is universal, that is it contains an isometric copy of all Polish metric spaces, and ultrahomogeneous, that is any isometry between finite subspaces of U extends to an isometry of U. Urysohn's metric space (and its isometry group) has been studied by topologists and descriptive set theorists for a plethora of reasons. In this talk, I will outline much of what is known on the model theory of Urysohn's metric space in the context of model theory for metric structures, which I will introduce at the beginning of my talk. I will discuss matters such as axiomatizability, quantifier elimination, independence relations, and definability.
Sept. 20, 2010
Alex Wilkie :
4 p.m. in SEO 612
Abstract
I will talk about definability theory for complex analytic functions and
an approach to Zilber's Conjecture on the quasi-minimality of the complex
exponential field. This will involve setting up certain pregeometries on
the complexes associated with existential definability in the complex
exponential field. I will then describe how an analytic continuation
preperty would yield a positive answer to Zilber's conjecture. The
remainder of the lectures will be an open ended discussion of how one
might establish this property.
Sept. 27, 2010
Alex Wilkie :
4 p.m. in SEO 612
Oct. 4, 2010
Alex Wilkie :
4 p.m. in SEO 612
Oct. 11, 2010
Alex Wilkie :
4 p.m. in SEO 612
Oct. 12, 2010
John Baldwin :
4 p.m. in SEO 612
Abstract
Translation from a sentence of $L_{\omega_1,\omega}$ to an associated atomic class is a key tool for the
study of categoricity in infinitary logic.
We compare the complexity of definition of model theoretic notions such as $\omega$-stability and
excellence for a sentence of $L_{\omega_1,\omega}$ and the associated atomic class. We show these properties are $\Sigma^1_2$ on the sentences and $\Pi^1_1$ on the classes. (Lower bounds have not been established in the atomic class case). In either case these properties are absolute. But the question of whether $\aleph_1$-categoricity is absolute remains open for either formulation. Connecting this study with more classical descriptive set theory we show that the class of models of a sentence of $L_{\omega_1,\omega}$ whose automorphism groups admit a complete left invariant metric is $\Pi^1_1$ but not $\Sigma^1_1$. Techniques from model theory, recursion theory and descriptive set theory are used. Much of this is joint work with David Marker.
Oct. 18, 2010
Alex Wilkie :
4 p.m. in SEO 612
Oct. 19, 2010
Lynn Scow :
4 p.m. in SEO 612
Oct. 25, 2010
Alex Wilkie :
4 p.m. in SEO 612
Oct. 26, 2010
Andrew Arana :
1 p.m. in SEO 636
Abstract
Over the years many mathematicians have voiced a preference for proofs
that stay "close" to the theorems being proved, avoiding "foreign",
"extraneous", or "remote" considerations. Such proofs have come to be
known as "pure". Examples abound, in geometry and number theory for
instance, and indeed one can see the Gödel phenomenon as an example as
well. In thinking about this preference two main questions arise: how
exactly can what is "foreign" to a statement be measured; and what reasons
are there for preferring pure proofs of a statement over impure proofs of
that same statement. In this talk we address both of these questions,
focusing on the first and indicating ways in which model theory bears on
its study.
Francois Loeser :
2:30 p.m. in SEO 636
Abstract
The Fundamental Lemma is a complicated combinatorial identity between integrals over local fields which plays a central feature in
the Langlands program in the theory of automorphic representations. A proof of the Fundamental Lemma was recently completed by Ngo Bau Chau for which he was awarded a Fields Medal in Hyderabad. His proof, which is geometric in nature, works for functions fields over finite fields. It has been proved previously by Waldspurger, using specific representation theory techniques, that the Fundamental Lemma over p-adic fields - which is the case more relevant for applications to number theory - would follow from the function field case, for p large enough. The aim of our talk is to explain how Waldspurger's result - and similar statements for various versions analogues of the Fundamental Lemma which are not covered by Waldspurger's result - follows at once from a general transfer result allowing to transfer identities between integrals depending on parameters from
functions fields over finite fields to p-adic fields, for large p, which we obtained in collaboration with Raf Cluckers using our theory of motivic integration in a definable setting. To achieve this, one has to carefully encode all the data appearing in the Fundamental Lemma in a definable way. This is joint work with Raf Cluckers and Tom Hales.
Alex Wilkie :
4 p.m. in SEO 636
Abstract
After some motivating remarks concerning Zilber's Conjecture on the quasi-minimality of the complex exponential field, I shall discuss the possible role of o-minimality in the study of complex exponentiation. I shall conclude with an observation on the definablility of algebro-logarithmic
functions which came to light during this investigation.
Nov. 2, 2010
Christian Rosendal :
4 p.m. in SEO 612
Abstract
Abstract: I will present the main ideas of the paper "The Structure of totally disconnected, locally compact groups" by G. Willis.
Relevant literature:
G. Willis: The structure of totally disconnected locally compact groups, Math. Ann. 300 (1994).
G. Willis: Further properties on the scale function on a totally disconnected group, J. Algebra 237 (2001)
Chatzidakis, Hrushovski: An invariant for difference field extensions, arXiv:0902.0844v3 [math.LO]
Nov. 8, 2010
Alex Wilkie :
4 p.m. in SEO 612
Nov. 9, 2010
Christian Rosendal :
4 p.m. in SEO 612
Abstract
Abstract: I will present the main ideas of the paper "The Structure of totally disconnected, locally compact groups" by G. Willis.
Relevant literature: G. Willis: The structure of totally disconnected locally compact groups, Math. Ann. 300 (1994).
G. Willis: Further properties on the scale function on a totally disconnected group, J. Algebra 237 (2001)
Chatzidakis, Hrushovski: An invariant for difference field extensions, arXiv:0902.0844v3 [math.LO]
Nov. 15, 2010
Alex Wilkie :
4 p.m. in SEO 612
Nov. 16, 2010
James Freitag :
4:30 p.m. in SEO 612
Abstract
This talk is a continuation of the past two talks on "The Structure of totally disconnected, locally compact groups" by G. Willis. In particular, we will talk about an application in the recent paper, "An invariant for difference field extensions." http://arxiv.org/PS_cache/arxiv/pdf/0902/0902.0844v3.pdf
We will warm up by discussing some classical invariants of difference field extensions. Then we will define the distance degree of a finite transcendence degree extension and explain what it has to do with the work G. Willis. If time permits, we will discuss results in the group theoretic setting suggested by results in the difference field setting.
Nov. 22, 2010
Alex Wilkie :
4 p.m. in SEO 612
Nov. 23, 2010
Cameron Hill :
4 p.m. in SEO 612
Abstract
This talk will consist of a sketch of the proof of a single main result linking geometric ideas from the first-order model theory of infinite structures with complexity-theoretic analyses of problems over classes of infinite structures. To remove any suspense, the statement of the theorem is as follows:
Theorem.
Let $K = fin[T G]$, where $T$ is a complete $k$-variable theory with infinitely many finite models up to isomorphism.
I. If $T$ is constructible, then $K$ is rosy.
II. $T$ is efficiently constructible if and only if $K$ is super-rosy.
Obviously, a great number of definitions are needed (regardless of the readers background, most likely) to make sense of these assertions. For the time being, it should be understood as a shadow of the "main current of first-order model theory" namely, Shelah's Classification theory. I take "efficiently constructible" meaning that models of $T$ can be efficiently recovered from elementary diagrams of subsets to be a reasonable substitute for "classifable" in the classical theory. We then seek a hierarchy of structural properties culminating in efficient constructibility in analogy with the stability-theoretic hierarchy, Stable $\supset$ Super-stable $\supset$ Classifable = Super-stable + NDOP. In the classical scenario, any non-trivial bound on the number of models of the theory in each cardinality imposes stability, which already supports the rudimentary notion of geometry known as non-forking independence. In the scenario of this study, the hypothesis of constructibility by an algorithm cursorily imitating that of an efficient algorithm in form (meaning, a practically-inationary program which isn't necessarily efficient) is sufficient to impose another rudimentary notion of geometry on the class of models in this case, known as independence in a rosy class; this is the content of I of the theorem. The further requirement of efficiency "polynomially-bounded running times" induces a further guarantee of good behavior in the geometry of independence, and the only if portion of II of the theorem amounts to just this fact. It turns out, then, that this additional tractability in the geometry gives enough purchase to devise an efficient algorithm, initially disguised as a weak model-theoretic coordinatization result, for the class of the theory's finite models.
Nov. 29, 2010
Alex Wilkie :
4 p.m. in SEO 612
Nov. 30, 2010
James Freitag :
4 p.m. in SEO 612
Abstract
This talk is a continuation of the past two talks on "The Structure of totally disconnected, locally compact groups" by G. Willis. In particular, we will talk about an application in the recent paper, "An invariant for difference field extensions." http://arxiv.org/PS_cache/arxiv/pdf/0902/0902.0844v3.pdf
I will explain what the group theoretic notion of tidy means when applied to automorphism groups of difference field extensions. Then we will discuss what sort of difference field extensions have tidy automorphism groups and explain why these are useful.
Jan. 25, 2011
Lynn Scow :
3 p.m. in SEO 612
Abstract
Shelah proved that any theory with the tree property has either TP1 or TP2.
I will survey some of the results from a recent paper by Professor Byunghan Kim and his student
Hyeung-Joon Kim, in which further properties around TP1 are explored. In fact, TP1 is equivalent to SOP2
for a theory, and so this work contributes to the effort to address the problem of whether SOP2 is a
distinct property from SOP1. In their paper, the authors furthermore develop a generalization of the
tree-indiscernibility first introduced in Dzamonja and Shelah's 2004 paper.
Feb. 1, 2011
Vincent Guingona :
3 p.m. in SEO 612
Abstract
In this talk, we discuss compression schemes and how they relate
to the model-theoretic notion of definability of types. This motivates
our definition of uniform definability of types over finite sets (UDTFS),
which is a generalization of the standard definability of types in stable
theories. We show that dp-minimal theories have UDTFS and discuss how
this impacts compression schemes. Finally, we discuss the notion of UDTFS
rank and show how it relates to VC-density and the dimension of a
compression scheme.
Feb. 8, 2011
John Baldwin :
3 p.m. in SEO 612
Abstract
Much of first order model theory studies the countable models of a
complete first order theory. When the sentence is taken to be in
$L_{\omega_1,\omega}$, `completeness' (in the sense of proving or
disproving every $L_{\omega_1,\omega}$-sense) implies
$\aleph_0$-categoricity. So while the Vaught conjecture for complete
first order theories has major results, the Vaught conjecture for
complete $L_{\omega_1,\omega}$ sentences is trivial. We study a
related question. Must a `complete' $\aleph_1$-categorical sentence
have at most countably many countable models? We provide several
examples where the answer is no under various notions of complete
(all of course weaker than above). These notions are attempts at a
more semantic definition of complete for AEC.
Feb. 15, 2011
Koushik Pal :
3 p.m. in SEO 612
Abstract
Fields are associated with different kinds of operators,
giving them the structure of difference fields, differential fields
and/or D-fields. Model theorists are interested in studying the
theories of such structures and axiomatizing them. Some of these
results can be generalized to fields with more than one operator.
Similar idea can be extended to valued fields as well, and one can
study valued fields with operators.
In this talk, I am going to give a small survey of nice model complete
theories of fields with different kinds of operators, leading up to
valued fields and my own work on valued difference fields.
Feb. 22, 2011
Jim Freitag :
3 p.m. in SEO 612
Abstract
We will investigate a notion of isogeny in superstable groups, generalizing (and
inspired by) the notion in algebraic groups and differential algebraic groups. As an
application, we will prove a uniqueness result for Baudisch's Jordan-Holder theorem
for superstable groups. If time allows, we will give applications to differential
algebraic groups.
March 1, 2011
Konstantin Slutsky :
3 p.m. in SEO 612
Abstract
Back in 1948 Graev gave a construction of two-sided invariant(or just tsi for short) metrics on the free groups
over metric spaces. These groups are the free objects in the category of groups with tsi metrics and Lipschitz homomorphism.
We will construct tsi metrics on the free products(possibly with amalgamation) of tsi groups. In very general terms we will
make one little step towards pushing concepts of geometric group theory to the continuous setting.
March 29, 2011
S. C. Song :
3 p.m. in SEO 612
Abstract
Ben Yaacov showed that types (over parameters) in the theory of
atomless random variable structures (ARV) correspond precisely to
(conditional) distributions in probability. Moreover, the logic topology on
types corresponds to the topology of weak convergence of distributions.
During this talk, I will define a new metric d* between types which is
equivalent to the usual d-metric in continuous model theory. Then I will
show the type space under the d*-metric is isometric to the Wasserstein
space, the space of distributions on [0, 1]^n, under the Wasserstein
distance. After this, I will show how the Kantorovich-Rubinstein duality
formula from optimal transport theory yields a formula for the d*-metric.
Finally, using results from continuous logic, I will generalize some results
in optimal transport theory.
April 12, 2011
Aleksandra Kwiatkowska :
3 p.m. in SEO 612
Abstract
A topological group G has ample generics if for every m the diagonal
conjugacy action of G on G^m has a comeager orbit. I will show that the group of
homeomorphisms of the Cantor set has ample generics. This answers a question of
Kechris and Rosendal. The main tool I use is the projective Fraisse theory, which is
a dualization of the Fraisse theory from model theory.
April 19, 2011
Chris Shaw :
3 p.m. in SEO 612
Abstract
Given that any o-minimal densely ordered group has full definable choice (namely,
definable Skolem functions and uniform elimination of imaginaries), it is a natural
question to ask whether this can be achieved in the weakly o-minimal setting. We
examine the case of a structure M' obtained by adding a new convex predicate to an
o-minimal structure M. If the new predicate is interpreted by a convex set bounded
on at least one side with an endpoint outside of M, then the resulting structure is
properly weakly o-minimal and has a weakly o-minimal theory. In this case, modulo
some definable elements, M' has definable Skolem functions present precisely when M'
is valuational.
April 21, 2011
Paul Larson :
3 p.m. in SEO 612
Abstract
Iterations of generic elementary embeddings with critical point $\omega_1$
are the fundamental construction underlying Woodin's
P$_{max}$ forcing, and they can be used to prove a number of $\Sigma_1$
absoluteness results with respect to the uncountable.
We will present proofs of the following using this method : (1) The
existence of a model for a statement of L$_{\omega_1, \omega}$(Q)
is forcing-absolute (2) If a PC_delta over L$_{\omega_1, \omega}$(aa) class
(forceably) has an uncountable model satisying uncountably
many types over a countable fragment of the language, then it has
2$^{\aleph_1}$ many uncountable models, each satisfying uncountably many
types over this
fragment, but pairwise satisfying just countably many in common. Time
permitting, we will discuss an extension of the Magidor-Malitz Theorem
using
this method.
April 26, 2011
Aaron Hill :
3 p.m. in SEO 612
Abstract
Two elements g and h of a Polish group G are topologically similar if for
every sequence (i_n) of integers, (g^{i_n}) converges to the identity iff (h^{i_n})
converges to the identity. We'll explore this notion in the group of invertible
measure-
preserving transformations, showing connections to mixing properties, centralizers,
and conjugacy classes of transformations.
June 24, 2011
Ichiro Ikeda :
11 a.m. in SEO 612
Abstract
Evans-Wong proved that the generic structure M_f defined by a
control function f always has NSOP_4. It is easily seen that the theory
of M_f is omega-categorical, and hence it has finite closure. I will show
that if the theory of a generic structure has finite closure, then it has
NSOP_4.
June 30, 2011
Ionannis Souldatos :
11 a.m. in SEO 612
Abstract
We will discuss the spectrum of complete sentences in $L_{\omega_1,\omega}$ of linearly ordered structures.
(A more complete abstract if available; email Baldwin).
David Marker :
9:30 a.m. in SEO 612
Abstract
I will begin by surveying some of the known results on uncountable models of counterexamples to Vaught's Conjecture for $L_{\omega_1,\omega}$-sentences and
then give a sketch of an unpublished result of Leo Harrington showing that counterexamples have models of unbounded Scott rank below $\omega_2$.
In particular this shows that a counterexample has at least $\aleph_2$ models of size $\aleph_1$.
Aug. 23, 2011
<> :
4 p.m. in SEO 1227
Aug. 30, 2011
Phillip Wesolek :
4 p.m. in SEO 1227
Abstract
This talk will outline the problem motivating Freer, Patel, and Ackerman's work on concentrated measures. We will first formally construct a measure concentrated on the isomorphism class of the random graph. Second, we will show this approach fails for the Henson graph.
Sept. 8, 2011
Julien Melleray :
3 p.m. in SEO 1227
Abstract
The space of actions of a given countable group on a fixed structure (e.g a separable Hilbert space) may often be endowed with a Polish topological structure. It is then natural to wonder whether one can describe which properties are generic (in the sense of Baire category). I'll discuss this problem, for countable abelian groups, in the settings of a separable Hilbert space and Urysohn's universal metric space.
This is joint work with T. Tsankov.
Sept. 13, 2011
James Freitag :
4 p.m. in SEO 427
Abstract
We will discuss how to apply stability theoretic techniques without letting model theory get in the way. There will be two variations on this theme:
1) Indecomposability theorems for groups in various categories.
2) Interpretable fields and definable Galois groups.
Sept. 20, 2011
Lynn Scow :
4 p.m. in SEO 427
Abstract
This continues the series of talks around invariant measures on classes of countable graphs. I will be talking about the Petrov-Vershik paper, "Uncountable Graphs and Invariant Measures on the Set of Universal Countable Graphs."
Sept. 27, 2011
Lynn Scow and Phil Wesolek :
4 p.m. in SEO 427
Abstract
We will talk about papers by Vaananen and Baldwin on the above topics, in view of Cherlin's question on finite triangle-free graphs.
Oct. 4, 2011
Donald Brower :
4 p.m. in SEO 427
Abstract
We exhibit a condition on indiscernible sequences which
almost seems to characterize the class of simple theories. The main
obstacle to getting a complete characterization will be presented.
Some other general properties of indiscernible sequences may be
covered if there is time.
Oct. 25, 2011
Todor Tsankov :
4 p.m. in SEO 427
Abstract
It is an interesting phenomenon that for many Polish groups, the algebraic structure "remembers" the topology.
This can be given many meanings; perhaps the strongest is the following automatic continuity property: every
homomorphism into a separable group is continuous. This property can be thought of as a strengthening of the
well-studied in model theory small index property: a Polish group has the small index property iff every homomorphism
into $S_\infty$ is continuous. The automatic continuity property was introduced by Kechris and Rosendal, who also
developed a technique for verifying it. More recently, their technique was generalized to the continuous setting by Ben
Yaacov, Berenstein and Melleray and that allowed a new kind of "two-step" proofs of automatic continuity, most notably
for the unitary group and the automorphism group of a standard probability space, examples which were inaccessible
by previous methods.
Oct. 27, 2011
Jay Williams :
3 p.m. in SEO 1227
Abstract
Descriptive set theory gives us a framework for analyzing the
relative complexity of quasi-orders (i.e. reflexive transitive relations)
arising in many areas of mathematics, such as Turing reducibility of sets of
natural numbers or embeddability of countable groups, using the notion of a
Borel reduction. I will discuss a special class of quasi-orders, the
countable Borel quasi-orders, and focus in particular on embeddability of
finitely-generated groups, answering a question of Louveau and Rosendal.
The ideas in this case will apply to the more general case of embeddability
of countable groups.
Nov. 8, 2011
Omar Leon Sanchez :
4 p.m. in SEO 427
Abstract
The talk
will be about differential-algebraic prolongations and how to use them to axiomatize
partial differentially closed fields. Also, after presenting generalized strongly
normal extensions in several derivations, we will see that each of such extensions
comes from a logarithmic differential equation on a group with an integrable section
of its prolongation.
Nov. 22, 2011
John Baldwin :
4 p.m. in SEO 427
Abstract
We describe techniques (ultralimits and omitting types theorem) for
constructing models of set theory with prescribed properties. Then we
use these properties of models of set theory to prove in ZFC theorems about
$L_{\omega_1,\omega}$ and $PC\Gamma(\aleph_0,\aleph_0)$ classes. E.g.
absoluteness of existence of a model in $\aleph_1$ (this proof by Larson), few
models in $\aleph_1$ implies small (in any fragment of
$L_{\omega_1,\omega}(aa))$ (Larson (earlier Keisler)) and for aec
(Baldwin/Larson), almost Galois $\omega$-stability and few models in $\aleph_1$
implies Galois $\omega$-stability (Baldwin/Larson/Shelah).
Nov. 29, 2011
Lynn Scow :
4 p.m. in SEO 427
Abstract
I will present on indiscernible trees and their applications. I say that a collection of parameters forms an indiscernible tree if, indexed by a structure that has a reduct to a partial tree order, it behaves like a general kind of indiscernible sequence. Certain properties of first-order theories, such as the tree property, behave well under taking certain indiscernible trees and not others. We will discuss these sorts of problems.
Jan. 17, 2012
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4 p.m. in SEO 427
Jan. 24, 2012
James Freitag :
4 p.m. in SEO 427
Abstract
First, motivation from differential algebraic geometry will be mentioned. Then we will prove Lascar's $\omega ^ \alpha $ theorem. Following this, if time allows, we will attempt to prove some additional restrictions on ranks involving regular types.
All of the necessary back round on geometric stability theory will be given. Basic familiarity with model theory will be assumed.
Feb. 7, 2012
Phillip Wesolek :
4 p.m. in SEO 427
Abstract
We discuss the first two sections of Laskowski's <i>a simpler axiomatization of the Shelah-Spencer almost sure theories</i>.
Feb. 14, 2012
Gabe Conant :
4 p.m. in SEO 427
Abstract
We discuss section three of Laskowski's "<i>A simpler axiomatization of the Shelah-Spencer almost sure theories</i>".
Feb. 21, 2012
Gabe Conant and Lynn Scow :
4 p.m. in SEO 427
Abstract
We continue our discussion of the paper.
Feb. 28, 2012
William Simmons :
4 p.m. in SEO 427
Abstract
In classical algebraic geometry, the role of compactness is played by the
property of completeness: if $V$ and $W$ are algebraic varieties, then $V$
is complete if the projection $V\times W \rightarrow W$ is a closed map
with respect to the Zariski topology. The fundamental theorem of
elimination theory asserts that projective varieties are complete. What
happens with differential varieties, i.e., solution sets of differential
equations over
differential fields? We discuss several approaches to the problem, with
our main focus being a positive quantifier elimination test of van den
Dries that was adapted to a differential valuative criterion by Pong.
March 6, 2012
Sarah Cotter :
4 p.m. in SEO 427
Abstract
A VC-minimal theory is one in which a directed family of definable sets generates all one-variable definable sets. We will look at some examples of VC-minimal theories, see how VC-minimality relates to other model-theoretic notions, and consider a specific class of VC-minimal theories which have additional nice structural properties. Finally, we'll outline the proof of a result characterizing forking over models in terms of definable types.
March 15, 2012
Tamvana Makuluni :
3 p.m. in SEO 1227
Abstract
We classify the complexity of the index set of uncountably categorical theories. We show that this index set surprisingly falls at the intermediate stage of being complete for intersections of $\Pi^0_2$-sets and $\Sigma^0_2$-sets. Work with Uri Andrews.
March 27, 2012
Fred Drueck :
4 p.m. in SEO 427
Abstract
We recall a theorem of Lessmann about producing
two-cardinal models based off the existence of a Vaughtian pair for an
AEC stable in $\aleph_0$ and discuss combinatorial difficulties to
extending this result to AECs with uncountable Löwenheim number. We
give an abstract framework for proving this result, which is an analog
of Vaught's Two Cardinal Theorem for AECs, which we call a
``superlimit'' and contrast this with Shelah's definition of
``superlimit''. We discuss also the difference between unions of
limit models being limit models and limit models being unique up to
isomorphism and discuss the possibility using Morasses to prove an
analog of Chang's gap-2 transfer theorem for AECs.
April 10, 2012
Dave Sahota :
4 p.m. in SEO 427
Abstract
Mayer showed a strong form of Vaught's Conjecture for o-minimal theories: if $T$ has fewer than $2^\omega$ countable models, then $T$ has $6^a3^b$ countable models for some natural numbers $a$ and $b$. We will show if $T$ has $2^\omega$ countable models, then if all $p(x)\in S_1(A)$ for any finite set $A$ are simple, then the class of countable models of T is Borel Reducible to the class of countable subsets of $2^\omega\times6$ by $M\mapsto\{(p,i)$ : the realizations of $p$ in $M$ are of form $i\}$; if any $p(x)\in S_1(A)$ is non-simple for some finite set $A$, then there is a finite set $B$ such that the class of countable models of $T$ over $B$ is Borel Complete.
April 17, 2012
Dana Bartosova :
4 p.m. in SEO 427
Abstract
I will introduce a construction by Balcar and Franek of the Boolean algebra of clopen sets of the universal minimal dynamical system for discrete groups. I will show how that generalizes to topological groups. I will talk about a couple of applications of this approach, e.g. to groups of automorphisms of uncountable structures using methods of Kechris, Pestov and Todorcevic connecting structural Ramsey theory and topological dynamics.
April 24, 2012
Koichiro Ikeda :
4 p.m. in SEO 427
Abstract
I would like to introduce two results with respect to generic structures. One is a theorem saying that there is no omega-categorical generic projective plane. The other is a theorem saying that if a generic structure has no cycle then the theory is not strictly superstable. Both of two are proved by some similar method, which is related to the rational approximations.
April 25, 2012
Anush Tserunnyan :
3 p.m. in SEO 427
Abstract
Consider a Borel action of a countable group G on a standard Borel space X. A countable Borel partition P of X is
called a generator if GP={gA: g in G, A in P} generates the Borel sigma-algebra of X.
Existence of such P of cardinality n is equivalent to the existence of a G-embedding of X into the shift n^G. For G=Z, the Kolmogorov-Sinai theorem
implies that finite generators don't exist in the presence of an invariant probability measure with infinite entropy.
It was asked by Weiss in the late 80s, whether the nonexistence of any invariant probability measure would guarantee the existence of a finite generator.
We show that the answer is positive in case X admits a sigma-compact topological realization (e.g. if X is a sigma-compact Polish G-space).
We also show that finite generators always exist in the context of Baire category thus answering a question
of Kechris. In fact, we show that if X is a Polish G-space having infinite orbits, then there is a
4-generator on an invariant comeager set.
June 13, 2012
Kota Takeuchi :
3 p.m. in SEO 427
Abstract
In model theory indiscernible sequences are very important objects. The unstability of a theory is witnessed by a sequence with an order. Moreover, we can retake the witness by an indiscernible sequence. This technique is useful for classifying unstable theories. For example, Tree Property is defined by the existence of a tree with a structure on the tree. To prove n-TP iff 2-TP, we use the existence of an indiscernible tree witnessing n-TP. In this talk we discuss when we have an indiscernible tree which satisfies a condition (like n-TP). The answer depends on the structure of the tree, but we will see that in many cases if the condition has a suitable subtree property then it is realized by an indiscernible tree.
Aug. 28, 2012
David Marker :
4 p.m. in SEO 427
Abstract
An integer part of a real closed field is a discretely
ordered subring where
every element of the field is within distance one of an element of the
ring. Sheperdson
first noticed that integer parts are models of a weak fragment of arithmetic.
Recently, D'Aquino, Knight and Starchenko studied the real closed fields where
the integer part is a model of Peano Arithmentic and gave a complete
classification in the
countable case. We will survey the subject and examine some phenomena
in the uncountable
case.
Sept. 11, 2012
Dima Sinapova :
4 p.m. in SEO 427
Abstract
The Singular Cardinal Problem is the project to completely describe the behavior of the operation $\kappa\mapsto 2^\kappa$ when $\kappa$ is singular. In this talk I will give some background and present forcing notions that are used to obtain consistency results. Then I will go over some recent developments in the subject.
Sept. 18, 2012
David Marker :
4 p.m. in SEO 427
Abstract
Antonio Montalban has proved (under the assumption of
projective determinacy) that
an $L_{\omega_1,\omega}$-sentence is a counterexample to Vaught's
Conjecture if and only
if there is a cone in the Turing degrees where every
$X$-hyperarithmetic model has an
$X$-recursive copy.
I will give background material and describe Montalban's results.
Sept. 25, 2012
Phillip Wesolek :
4 p.m. in SEO 427
Abstract
We begin the discussion of ``Invariant measures concentrated on countable structures'', by Ackerman-Freer-Patel . We discuss Borel L-structures and show how to obtain invariant measures from such structures.
Oct. 2, 2012
Gabe Conant :
4 p.m. in SEO 427
Abstract
We continue the discussion of ``Invariant measures concentrated on countable structures'', by Ackerman-Freer-Patel . We show how to obtain a Borel L-structure witnessing the pithy $\Pi_2$ expansion of the Scott sentence of a countable L-structure with trivial group theoretic definable closure.
Oct. 9, 2012
Marcin Sabok :
4 p.m. in SEO 427
Abstract
I will discuss some recent developments in canonical Ramsey theory in
the context of descriptive set theory. I will survey canonization
results for various classes of equivalence relations. I will show that
hypersmooth equivalence relations canonize to three equivalences: two
trivial ones and $E_1$. I will also show that equivalence relations
reducible to $E_2$ also canonize to three equivalences: the two
trivial ones and $E_2$.
This is part of a joint forthcoming book with Vladimir Kanovei and
Jindra Zapletal.
Patrick Reynolds :
2 p.m. in SEO 427
Abstract
For F a free group and $\Phi \in$ Out(F) an element of infinite
order, a fruitful approach for studying the structure of $\Phi$ is to
construct from $\Phi$ an isometric action of F on an $\mathbb{R}$-tree
$T_{\Phi}$ that is preserved by $\Phi$ in an appropriate sense. The idea is
that certain aspects of the structure of $\Phi$ are reflected in the
structure of $T_{\Phi}$. The typical procedure for constructing $T_{\Phi}$ is
to use the equivariant Gromov topology on the space of F-spaces. The
point of this talk is to give a simple construction of $T_{\Phi}$ from the
point of view of non-standard analysis. We will further explain that the
non-standard picture makes clear how to construct a more sensitive
invariant, and we will give a discussion of the advantages of this new
invariant. Relevant group theory and non-standard analysis background
will be given, so the talk should be generally accessible.
Oct. 16, 2012
Bektur Baizhanov :
4 p.m. in SEO 427
Abstract
A well-developed technique created to study stable theories (M. Morley, S. Shelah) is applied in dealing with a class of theories with definable linear order. We introduce the notion of an o-stable theory, which generalizes the concepts of o-minimality, of weak o-minimality, and of quasi-o-minimality. It is proved that o-stable theories are dependent, but they do not exhaust the class of dependent theories with definable linear order, and that every linear order is o-superstable.
Results obtained together with Viktor Verbovsky
Oct. 30, 2012
Dima Sinapova :
4 p.m. in SEO 427
Abstract
Investigating which forcings preserve stationary sets is a major topic in modern set theory. I will survey when we can generally expect stationary sets to be preserved by forcing. I will also give some examples of how forcing can destroy stationary sets. Finally, I will go over a recent result.
Nov. 6, 2012
Kostas Beros :
4 p.m. in SEO 427
Abstract
Given a topological group G and a family C of subgroups of G
define a "universal C subgroup of G" to be a subgroup K in the family C such
that every subgroup H in C is a continuous homomorphic pre-image of K.
My recent work has shown that the countable power of a locally compact
Polish group has universal compactly generated and K-sigma subgroups and
that the countable power of an arbitrary Polish group has a universal
analytic subgroup. For reasons I will explain in my talk, compactly
generated, K-sigma and analytic subgroups are natural classes to work with
in this context and share some useful properties.
I will give some motivation for studying universal subgroups, summarize my
results so far (including the ones above) and prove at least one of these
results in the case of the Baer-Specker group.
Nov. 13, 2012
Scott Cramer :
4 p.m. in SEO 427
Abstract
The structure $L(V_{\lambda+1})$ was first studied by Woodin to prove the consistency of $AD^{L(R)}$ from large cardinals. He later showed that many of the same structural properties of $L(R)$ under determinacy hold for $L(V_{\lambda+1})$ under large cardinals. Laver first introduced the tool of inverse limits in this context to tackle the problem of reflecting large cardinals at this level. In this talk we will extend Laver's results on inverse limit reflection, and then use this technique to analyze further the structure of $L(V_{\lambda+1})$ and its relationship to models of determinacy.
Nov. 20, 2012
John Baldwin :
4 p.m. in SEO 427
Abstract
I will discuss several applications of a method of Shelah to build a
model in the continuum from a countable model satisfying certain
geometric conditions. In particular, this provides a streamlined
argument for the first part of the Ackerman, Freer, and Patel paper
discussed earlier in the seminar.
Theorem. Let $B$ be a countable model with a
totally trivial closure operation. Then there is an uncountable Borel
model which `strongly witnesses' Th($M$).
Another application is to the notion recently introduced by Shelah,
which I will call pseudoclosure, pcl.
Theorem. If there is a quasiminimal $M \in \mathbf{K}_T$,
where pcl satisfies exchange, with cardinality $\aleph_1$, then there is an $N \in \mathbf{K}_T$ with cardinality $2^{\aleph_0}$.
A key point that I will mention in the seminar is the reduction from
a complete sentence of $L_{\omega_1,\omega}$ to the class of atomic
models of an associated first order theory. The details are in
section 6.1 of my monograph, summarized in Theorem 6.1.8.
http://homepages.math.uic.edu/~jbaldwin/pub/AEClec.pdf
Nov. 27, 2012
Isaac Goldbring :
4 p.m. in SEO 427
Abstract
In this talk, we will show that the theory of tracial von Neumann algebras does not have a model companion. In addition, we will show that a positive solution to the Connes Embedding Problem implies that there is no model complete theory of tracial von Neumann algebras. All functional analytic notions and most model-theoretic notions will be defined. This is joint work with Bradd Hart and Thomas Sinclair.
Dec. 4, 2012
Phillip Wesolek :
4 p.m. in SEO 427
Abstract
We give a proof of the following theorem: If $G$ is a locally compact, totally disconnected Polish group with a dense conjugacy class $g^G$, then $g^G$ is Haar null. This answers a question of Kechris and Rosendal in the negative.
Feb. 5, 2013
Uri Andrews :
4 p.m. in SEO 427
Abstract
The (degree) spectrum of a theory is the set of Turing degrees which compute a model of the theory. I'll discuss several new theorems relating theory spectra to model theoretic properties. (Various parts of the talk joint with Joseph S. Miller; Julia Knight; or Mingzhong Cai, David Diamondstone, Steffen Lempp, Joseph S. Miller)
Feb. 12, 2013
Lynn Scow :
4 p.m. in SEO 427
Abstract
We survey some cases where generalized indiscernible sets
are applied to obtain many-models results. In particular, we present
some details from the Laskowski and Shelah 2003 paper, ``Karp
complexity and classes with the independence property.'' If time
permits, we hope to present Ziegler's argument in his 1988
``Stabilit\"{a}tstheorie'' notes that Shelah tree-indiscernibles have
the modeling property.
Feb. 19, 2013
Isaac Goldbring :
4 p.m. in SEO 427
Abstract
We discuss the relationship between fields of transseries and residue fields obtained from convex subrings of nonstandard extensions of the real numbers.
Feb. 26, 2013
Alex Rennet :
4 p.m. in SEO 427
Abstract
For a fixed language L, the first-order L-theory of o-minimality is the set of those L-sentences true in all o-minimal L-structures. It follows from a classical model-theoretic result that a model of this theory is either o-minimal or an elementary substructure of an ultraproduct of o-minimal L-structures. For most languages, the latter kind of model will not in general be o-minimal; we call these structures pseudo-o-minimal.
In this talk, I will discuss how the study of pseudo-o-minimality fits in to the ongoing project of classifying the tame weakenings of o-minimality. My main focus will be on the recent question of whether for certain fixed languages L, the first-order L-theory of o-minimality is recursively axiomatizable. I will show that it is not whenever L extends the language of ordered fields by at least one new predicate or function symbol. With the time remaining, I will outline some of what is known about the relative tameness of pseudo-o-minimal structures, mention some open problems in the area, and discuss some potential applications.
March 5, 2013
Howard Becker :
4 p.m. in SEO 427
Abstract
The following question in computable model theory is
open: Does there exist a hyperarithmetic class of computable
structures with exactly one non-hyperarithmetic isomorphism-type?
Given any oracle a in 2^omega, we can ask the same question
relativized to a. A negative answer for every a implies Vaught's
conjecture.
March 12, 2013
Sergey Sudoplatov :
4 p.m. in SEO 427
Abstract
We present a solution of the Lachlan problem on the existence of
stable Ehrenfeucht theories, and syntactic generic constructions as
keys for the classification of countable models of complete
theories with respect to two basic characteristics: Rudin-Keisler
preorders and distribution functions for limit models. As a result
we obtain a classification of countable models (up to the
continuum hypothesis) for the class of Ehrenfeucht theories, for
the class of small theories, and for the class of theories with
continuum many types (the last one is joint with Roman Popkov).
April 4, 2013
Jindra Zapletal :
3 p.m. in SEO 427
Abstract
For analytic equivalence relations E, F on Polish spaces, say that E is F-generically ergodic if for every Borel homomorphism from E to F there is an F-equivalence class whose preimage is comeager. Similarly define F-measure ergodicity. I will outline a new technology for proving such ergodicity results, starting with a forcing restatement of Hjorth's turbulence. A sample result: the $c_0$ equivalence is F-measure-ergodic for every K-sigma equivalence relation F.
April 9, 2013
Slawomir Solecki :
4 p.m. in SEO 427
Abstract
I will present an abstract approach to finite Ramsey theory, which reveals the formal algebraic
structure underlying results of that theory and which yields most of the results of the theory. I will formulate
within this approach an abstract pigeonhole principle and an abstract Ramsey condition, and state a theorem
that the pigeonhole principle implies the Ramsey condition. I will illustrate how the general approach is applied
on new concrete examples of Ramsey statements---a common generalization of Deuber's and Jasinski's
Ramsey statements for trees, the self-dual Ramsey statement, and a Ramsey statement for 1-Lipschitz functions.
April 16, 2013
James Freitag :
4 p.m. in SEO 427
Abstract
We will introduce differential algebraic varieties and the notion of completeness in this category. We will discuss how to determine if a differential algebraic variety is complete and potential applications of the fact. In the case of partial differential equations, we will discuss how completeness may be easily used to give necessary and sufficient conditions for the linear dependence (and more general notions of dependence) for elements in differential fields.
April 25, 2013
Jesse Johnson :
3 p.m. in SEO 427
Abstract
We will introduce some basic notions of $\alpha$-recursion as applied to computable structure theory. We will give a few easy examples of ``computable" structures and "computably categorical" structures. Using these notions, we will give a computability-theoretic analysis of quasiminimal-excellent classes. We show that for any quasiminimal-excellent class (with infinite-dimensional models) and any successor $\kappa^+ \geq \aleph_1$, the member of dimension $\kappa^+$ has a computable copy and is relatively $\Delta^0_2$-categorical. We then give precise conditions under which the member of size $\kappa^+$ is relatively-computably categorical.
April 30, 2013
Grigor Sargsyan :
4 p.m. in SEO 427
Abstract
We will state a covering conjecture and explain how it can be used to derive strength from PFA and other statements.
Aug. 27, 2013
Isaac Goldbring :
4 p.m. in SEO 427
Abstract
Erdös conjectured that any set A of natural numbers of <i>positive lower density</i> necessarily contains a sumset B+C, where B and C are infinite sets of natural numbers. In this talk, I will show how techniques from nonstandard analysis can be used to make progress on this conjecture. In particular, we settle this conjecture when A has <i> Banach density</i> exceeding 1/2 and use this result to prove a "one-translate" version of the conjecture for arbitrary A of positive Banach density. All necessary notions from combinatorial number theory and nonstandard analysis will be introduced. This work is joint with Mauro DiNasso, Renling Jin, Steven Leth, Martino Lupini, and Karl Mahlburg and was partially done during our Squares Week at AIM.
Sept. 3, 2013
Dave Marker :
4 p.m. in SEO 427
Abstract
In an earlier seminar I showed that, assuming
$\diamondsuit$, there is a family of $2^{\aleph_1}$ recursively saturated
models of Peano Arithmetic with the same theory and standard system
but non-isomorphic real closures. I will give a new proof of this without
the assumption of $\diamondsuit$. This is joint work with Jim Schmerl and
Charlie Steinhorn.
Sept. 24, 2013
Gabriel Conant :
4 p.m. in SEO 427
Abstract
In classical model theory, the strong order property hierarchy is used to stratify the non-simple theories without the strict order property. This is the first of two talks in which we discuss the (non-simple) theory of the Urysohn sphere, viewed as a metric structure in continuous model theory. We will define the continuous version of the strong order property, place the Urysohn sphere in this hierarchy, and finally give a characterization of forking independence.
Oct. 1, 2013
Caroline Terry :
4 p.m. in SEO 427
Abstract
In classical model theory, the strong order property hierarchy is used to stratify the non-simple theories without the strict order property. This is the first of two talks in which we discuss the (non-simple) theory of the Urysohn sphere, viewed as a metric structure in continuous model theory. We will define the continuous version of the strong order property, place the Urysohn sphere in this hierarchy, and finally give a characterization of forking independence.
Oct. 8, 2013
Spencer Unger :
4 p.m. in SEO 427
Abstract
In this talk we focus on generalizing two theorems of Mitchell, but in different directions. The theorems of Mitchell are
Thm 1: The tree property at $\aleph_2$ is equiconsistent with the existence of weakly compact cardinal.
Thm 2: The failure of weak $\square_{\aleph_1}$ is equiconsistent with a Mahlo cardinal.
We develop some of the tools needed to prove the following two theorems.
Thm1': Assuming there is a weakly compact cardinal it is consistent that the tree property holds at $\aleph_2$ and the continuum is larger than $\aleph_2$.
Thm2': The failure of weak $\square_{\aleph_n}$ for all n > 0 is equiconsistent with infinitely many Mahlo cardinals.
Oct. 15, 2013
Chris Miller :
4 p.m. in SEO 427
Abstract
There are expansions of dense linear orders by open sets (of arbitrary arities) such that all of the following hold:
---Every definable set is a boolean combination of existentially definable sets.
---Some definable sets are not existentially definable.
---Some projections of closed bounded definable sets are somewhere both dense and codense.
---There is a unique maximal reduct having the property that every unary definable set either has interior or is nowhere dense. It properly expands the underlying order, yet is still rather trivial.
At least some of these structures come up naturally in model theory. For example, if G is a generic predicate for the real field, then the expansion of G by the G-traces of all semialgebraic open sets is such a structure, which moreover is interdefinable with the structure induced on G in (R,+,x,G).
(Recent joint work with A. Dolich and C. Steinhorn, but any errors are
mine.)
Oct. 29, 2013
Martin Bays :
4 p.m. in SEO 427
Abstract
As part of his programme to tackle the model theory of complex exponentation, Zilber (2002) obtained a categoricity result for the structure of the exponential map in the "Lie algebra" language $<\mathbb C;+> --> <\mathbb C^*;+,*>$, where the domain has only linear structure.
I will present an abstract version of this result, where $\mathbb C^*$ is
replaced by an almost arbitrary commutative finite Morley rank
group, for example by a semiabelian variety in arbitrary
characteristic.
This is work-in-preparation with Bradd Hart and Anand Pillay.
Nov. 5, 2013
Martin Zeman :
4 p.m. in SEO 427
Abstract
The talk focuses on ideals on $\omega_2$ and associated generic embeddings. Such generic embeddings arise in set theory naturally, and their existence has important impact on combinatorics at small cardinals and on the structure of sets of reals. For this reason, some set theorists view them as possible alternatives for large cardinal axioms. One of the most well-known open problem concerning ideals on $\omega_2$ is the saturation of the non-stationary ideal on $\omega_2$ restricted to cof$(\omega_1)$. We formulate a weakening of this property in terms of self-generic structures which is interesting on its own, prove it is consistent relative to large cardinals, and show it has large cardinal strength. This is a joint work with Sean Cox.
Nov. 12, 2013
Maryanthe Malliaris :
4 p.m. in SEO 427
Abstract
The talk will be about the framework of "cofinality spectrum problems," introduced in a recent paper of Malliaris and Shelah (arxiv:1208.5424).
Nov. 19, 2013
Dima Sinapova :
4 p.m. in SEO 427
Abstract
We investigate the relationship between square properties and the failure of the Singular Cardinal Hypothesis (SCH). We will focus on models where SCH fails at $\aleph_\omega$ and models when SCH fails at $\kappa$ while GCH holds below $\kappa$. We will also talk about very good scales and how they interact with both square properties and SCH.
Dec. 3, 2013
Victor Ocasio :
4 p.m. in SEO 427
Abstract
The class of Real Closed Fields (RCF) is known to have very nice model theoretic properties, among them o-minimality and quantifier elimination. In our work, we consider some non- elementary subclasses of RCF and explore their computability theoretic properties. We locate the class of Archimedean Real Closed Fields using Turing computable embeddings (an analog of Borel embeddings) and compare it with other non-elementary first order subclasses of RCF. We also explore relative categoricity and show that under some conditions one can obtain a sharp result on the complexity of the relative categoricity of a real closed field that is constructed using a linear order as an oracle.
Dec. 10, 2013
Ioannis Souldatis :
3:30 p.m. in SEO 427
Abstract
An $L_{\omega_1,\omega}$ sentence $\phi$ characterizes a cardinal $\kappa$, if $\phi$ has models in all powers up to $\kappa$, but not in $\kappa^+$.
During the talk we will focus on the results and the construction in Hjorth's [1], and state some extensions of these results, as well as questions that remain open.
[1] Hjorth, G.
Knight's model, its automorphism group, and characterizing the uncountable cardinals
J. Math. Log., 2002, 2, 113-144
Jan. 21, 2014
Christian Rosendal :
4 p.m. in SEO 427
Abstract
Large scale geometry on the one hand originates in Banach space theory, notably by work of Enflo and Ribe, while in the setting of countable discrete groups is mainly an invention due to Gromov. The fundamental observation here is that the word metric on a finitely generated group is left-invariant and, while not unique since it involves a choice of finite generating set, it is independent of the generating set up to quasi-isometry. In the setting of locally compact second countable groups the situation is essentially equivalent since, by a result of Struble, every compactly generated G group admits a compatible left-invariant proper metric quasi-isometric to the word metric of a compact generating set.
However, if G is a metrisable topological group, though G admits many left-invariant metrics, it is not clear that there should be any manner of defining a unique quasi-isometry type since there is no canonical generating set for G. Nevertheless, we show how to overcome this problem by considering a notion of "metric compactness", which we enable us to compute the intrinsic quasi-isometric type of various such groups. We also present associated results for metrically proper affine actions on Banach spaces and discuss how our theory plays out for automorphism groups of countable first-order structures.
Jan. 28, 2014
Christian Rosendal :
4 p.m. in SEO 427
Abstract
Large scale geometry on the one hand originates in Banach space theory, notably by work of Enflo and Ribe, while in the setting of countable discrete groups is mainly an invention due to Gromov. The fundamental observation here is that the word metric on a finitely generated group is left-invariant and, while not unique since it involves a choice of finite generating set, it is independent of the generating set up to quasi-isometry. In the setting of locally compact second countable groups the situation is essentially equivalent since, by a result of Struble, every compactly generated G group admits a compatible left-invariant proper metric quasi-isometric to the word metric of a compact generating set.
However, if G is a metrisable topological group, though G admits many left-invariant metrics, it is not clear that there should be any manner of defining a unique quasi-isometry type since there is no canonical generating set for G. Nevertheless, we show how to overcome this problem by considering a notion of "metric compactness", which we enable us to compute the intrinsic quasi-isometric type of various such groups. We also present associated results for metrically proper affine actions on Banach spaces and discuss how our theory plays out for automorphism groups of countable first-order structures.
Feb. 4, 2014
Phillip Wesolek :
4 p.m. in SEO 427
Abstract
The class of constructible totally disconnected locally compact (t.d.l.c.) Polish groups is the collection of t.d.l.c. Polish groups built from profinite and discrete groups via group extension and countable increasing union. These groups appear often in the study of t.d.l.c. Polish groups. We show this class satisfies surprisingly robust closure properties. We go on to give an application to the study of $p$-adic Lie groups. In particular, we show every $p$-adic Lie group decomposes into constructible and topologically simple groups via group extensions. This result is analogous to the solvable by semi-simple decomposition for connected Lie groups. Time permitting, we discuss a second application to a question of Gao's on surjectively universal t.d.l.c. Polish groups.
Feb. 11, 2014
Gregory Igusa :
4 p.m. in SEO 427
Abstract
We say that a real, $A$, is generically computable if there is a partial computable function $\phi$ which correctly computes the majority of the bits of $A$, and never gives any incorrect outputs. (So dom$(\phi)$ is a subset of $\omega$ with asymptotic density 1.) The terminology and motivation for this are derived from recent work in complexity theory which studies the ``generic-case complexity" of a problem which might, in principle, be much easier in the generic case than in the worst case.
If we wish to relativize generic computability to study its degree structure, we are forced to work with uncountable collections of partial oracles for each real: any computation that halts on density 1 is an acceptable generic computation, and so any oracle that answers density-1 many questions must be an acceptable generic oracle. This produces a $\bf{\Pi}^1_1$-complete reducibility that is somewhat difficult to work with.
The Turing degrees embed naturally into the generic degrees, and this provides a very useful perspective from which to view the generic degrees. Given a generic degree, we may consider the Turing ideal or filter of Turing degrees that embed below or above it. We present some results about what sorts of ideals and filters can be realized in this manner, and we also provide a characterization of the hyperarithmetic sets in terms of generic reduction and coarse computability, a closely-related notion of computability.
March 4, 2014
Anush Tserunyan :
4 p.m. in SEO 427
Abstract
A major theme in ergodic Ramsey theory and multiplicative combinatorics is proving multiple recurrence results for certian doubly recurrent (mixing) actions of semigroups. The amplification of double to multiple recurrence is usually done using a so-called van der Corput difference (ratio) lemma for a suitable filter on the semigroup. Particular instances of this lemma (for concrete filters) have been known proven before (by Furstenberg, Bergelson-McCutcheon, and others), with a different proof for each filter. We define a general class of filters on semigroups, which includes all of the filters for which the van der Corput lemma was known. For the filters in this class (call them Delta-filters), we prove a Ramsey theorem related to labeling edges between the semigroup elements with their ratios. An application of this theorem yields a van der Corput lemma for Delta-filters, generalizing all its previous instances.
March 11, 2014
Henry Towsner :
4 p.m. in SEO 427
Abstract
The many equivalent characterizations of quasirandomness for graphs have been extensively studied. When generalized to hypergraphs, these notions split into a partially ordered family of distinct notions, recently shown by Lenz and Mubayi to not even be linearly ordered.
The ultraproduct setting, equipped with Loeb measure, turns out to be a natural place to examine these notions; in this setting, the different notions of quasirandomness for hypergraphs match up to certain natural algebras of definable sets. Considering all possible variations of these algebras includes all the notions studied so far, and introduces a few new notions. For some characterizations of graph randomness we are able to produce a generalization corresponding to each possible algebra. In particular, for each algebra we identify the class of hypergraphs which appear with the "correct" frequency in any hypergraph random for that algebra.
March 18, 2014
Philipp Hieronymi :
4 p.m. in SEO 427
Abstract
Let R be the set of real numbers and let Z be the set of integers. The theory (R,<,+,Z,aZ) is decidable if a is quadratic. If a is the golden ratio, (R,<,+,Z,aZ) defines multiplication by a. The results are established by using the Ostrowski representation of a real number based on the continued fraction expansions of a to define the above structures in monadic second order logic of one successor.
April 8, 2014
Isaac Goldbring :
4 p.m. in SEO 427
Abstract
The Connes Embedding Problem (CEP) asks whether or not every separable II_1 factor embeds into an ultrapower of the hyperfinite II_1 factor R. In last year's seminar, I showed how a negative solution to the CEP would follow from showing that Th(R) is model complete. In this talk, I will show that Th(R) is not model complete, destroying the aforementioned plan to settle CEP. However, I will then explain a new model-theoretic connection with CEP, namely that CEP is equivalent to every type II_1 algebra having a computable universal theory. The first part of the talk is part of joint work with Ilijas Farah, Bradd Hart, and David Sherman while the second part of the talk is joint work with Bradd Hart.
April 15, 2014
Dima Sinapova :
4 p.m. in SEO 427
Abstract
It is difficult to avoid the weaker square principles at successors of singulars. Doing so requires large cardinals. It is especially difficult to obtain these failures while also violating the Singular Cardinal Hypothesis (SCH). The latter is usually achieved by singularizing some large cardinal. I will discuss the impact of singularizing cardinals on square properties. This is joint work with M. Magidor
April 22, 2014
Itay Neeman :
4 p.m. in SEO 427
Abstract
The tree property at $\kappa$ asserts that every tree of height $\kappa$
with levels of sizes less than $\kappa$ has a cofinal branch. At
$\kappa=\aleph_0$ and $\kappa=\aleph_1$ the property and its negation
(respectively) are well known classical results. We discuss methods to
obtain the property at other cardinals, where it can be viewed as a
delicate remnant of large cardinal strength.
April 29, 2014
Slawomir Solecki :
4 p.m. in SEO 427
Abstract
Galvin--Glazer--type results are an important part of infinite Ramsey theory. A number of them are known. I will describe a setting that makes it possible to state a general such theorem, which I will state. It leads to a new kind of problems in dynamics of semigroups with, usually finite, semigroups acting on compact semigroups via continuous homomorphisms.
Sept. 2, 2014
Will Boney :
4 p.m. in SEO 427
Abstract
Tameness is a locality property of Galois types in AECs. Since its isolation by Grossberg and VanDieren 10 years ago, it has been used to prove new results (upward categoricity transfer, stability transfer) and replace set-theoretic hypotheses (existence of independence notions). In this talk, we will outline the basic definitions, summarize some key results, and discuss some open questions related to tameness.
Sept. 9, 2014
Thomas Sinclair :
4 p.m. in SEO 427
Abstract
A C*-algebra A is said to be existentially closed if, roughly, every set of equations involving norms of noncommutative *-polynomials which has a solution in B(H) has a sequence of approximate solutions in A. A basic result in continuous logic shows that every separable C*-algebra is contained in a separable, existentially closed C*-algebra. In this talk I will survey some basic properties of existentially closed C*-algebras. In particular I will describe how existential closure is deeply connected to several open problems in C*-algebras such as Kirchberg's problem on whether every separable C*-algebra embeds in an ultrapower of the Cuntz algebra O_2, as well as Kirchberg's C*-algebraic reformulation of of Connes' embedding problem. This talk is based on joint work with Isaac Goldbring.
Sept. 16, 2014
Spencer Unger :
4 p.m. in SEO 427
Abstract
The tree property arises as the generalization of Konig's infinity lemma to an uncountable cardinal. The existence of an uncountable cardinal with the tree property has axiomatic strength beyond the axioms of ZFC. Indeed a theorem of Mitchell shows that the theory ZFC + ``omega_2 has the tree property" is consistent if and only if the theory ZFC + ``There is a weakly compact cardinal" is consistent. In the context of Mitchell's theorem, we can ask an old question in set theory: Is it consistent that every regular cardinal greater than aleph_1 has the tree property? In this talk we will survey the best known partial results towards a positive answer to this question.
Sept. 23, 2014
Joseph Zielinski :
4 p.m. in SEO 427
Abstract
We consider the analysis of classification problems in the context of Borel reducibility and outline a proof that the complexity of the homeomorphism relation between compact metric spaces coincides, in this way, with that of the complete orbit equivalence relation of Polish group actions.
Oct. 7, 2014
Kostya Slutskyy :
4 p.m. in SEO 427
Abstract
A cross-section of a Borel flow is a Borel set that has
countable intersection with each orbit of the flow. We shall be
interested in constructing cross-sections with a prescribed set of
possible distances between adjacent points within orbits. The main
result of the talk is that given any two rationally independent
positive reals and a free Borel flow one can always find a
cross-section with distances between adjacent points being only these
two real numbers.
We shall give an overview of the subject from both ergodic theoretical
and descriptive points of view and an application of the above result
to orbit equivalence of flows will be presented.
Oct. 14, 2014
Erik Walsberg :
4 p.m. in SEO 427
Abstract
I will discuss Gromov-Hausdorff limits in the o-minimal
setting. Joint work with Ehud Hrushovski.
Oct. 16, 2014
James Freitag :
1 p.m. in SEO 636
Abstract
Take $ \alpha \in GL_2$ and a complex number $a$. There are at most $36^7$ complex numbers $b$ such that the elliptic curves $E_a$ and $E_b$ are isogenous and $E_ {\alpha (a)}$ and $E_ {\alpha (b)} $ are isogenous. Proving this fact along with an effective form of a special case of the Zilber-Pink conjecture uses input from model theory, differential algebra, and diophantine geometry. We will describe the proof and partial generalizations to various moduli spaces of abelian varieties.
Oct. 28, 2014
C. Ward Henson, Krzysztof Krupiński and Ramin Takloo-Bighash :
1 p.m. in SEO 636
Abstract
There will be no logic seminar. Instead, we will be hosting Midwest Model Theory Day http://math.wisc.edu/~andrews/MWMTD8.html.
Nov. 11, 2014
Isaac Goldbring :
4 p.m. in SEO 427
Abstract
A common theme in combinatorial number theory is to deduce structure in subsets of the natural numbers that are not small with respect to some density. Perhaps the most famous example of such a result is Szemeredi's Theorem, which states that if $A$ is a subset of the natural numbers with positive upper density, then $A$ contains arbitrarily long arithmetic progressions.
A motivating example for this talk will be a theorem of Renling Jin, which states that if $A$ and $B$ are subsets of the natural numbers with positive Banach density, then $A+B$ is piecewise syndetic, meaning that there is a natural number $k$ such that $A+B+[0,k]$ contains arbitrarily long intervals. Jin's proof uses nonstandard analysis, and, in particular, the notion of Loeb measure.
In this talk, we will focus on recent applications of nonstandard analysis to combinatorial number theory which rely not on the Loeb measure spaces, but rather on certain quotients of them called monad measure spaces. After defining the monad measure spaces, we will show how a Lebesgue Density Theorem for these spaces easily yields Jin's theorem. In addition, I will explain how, with a little more effort, one can even deduce certain quantitative versions of Jin's theorem.
I will end the talk with recent applications of a multiplicative (or logarithmic) version of the monad measure space construction, which we use to obtain approximate geo-arithmetic structure in sets of positive logarithmic density.
Much of the work presented in this talk is joint work with Mauro di Nasso, Renling Jin, Steven Leth, Martino Lupini, and Karl Mahlburg.
Nov. 18, 2014
Martino Lupini :
4 p.m. in SEO 427
Abstract
Working in the framework of Fraisse theory for metric structures developed by Ben Yaacov, we show that the noncommutative Gurarij space introduced by Oikhberg can be characterized as the Fraisse limit of the class of 1-exact finite-dimensional operator spaces. As a consequence we deduce that such an operator space is unique, homogeneous, and universal for separable 1-exact operator spaces.
Nov. 25, 2014
John Baldwin :
4 p.m. in SEO 427
Abstract
I will discuss some aspects of continuing joint work with Friedman,
Koerwein, Larson, Laskowski, and Shelah. This work uses forcing
techniques to prove model theoretic results in ZFC. Force a model
theoretic result to be consistent by a tool such as Martin's axiom,
collapsing cardinals or a specific forcing with high model theoretic
content. Then use iterated elementary embeddings of the model of set
theory to show the model theoretic result is absolute between V and a
well-chosen model. Deduce it holds in ZFC. Applications include
various extensions of results for $L_{\omega_1,\omega}$ to
analytically presented AEC, a new proof of Harrington's theorem on
Scott rank of counterexamples to Vaught's conjecture and the
development of a new notion of algebraic closure for
$L_{\omega_1,\omega}$ that better explains $\aleph_1$-categoricity.
I will briefly contrast this with results about the characterization
of cardinals. The circle is closed by concluding (95\% now) from
arguments of the first sort that if a sentence of
$L{\omega_1,\omega}$ characterizes a cardinal below the continuum
then it has $2^{\aleph_1}$ models in $\aleph_1$.
Dec. 2, 2014
Pavol Zlatos :
4 p.m. in SEO 427
Abstract
Using the ideas of E. I. Gordon [Go1], [Go2] we present an approach, based on nonstan-
dard analysis (NSA), to simultaneous approximation of locally compact abelian (LCA)
groups and their duals by nite abelian groups, as well as to approximation of the Fourier
transforms on various functional spaces over them by the discrete Fourier transform. In
2012 we proved the three Gordon's Conjectures (GC1{3) which were open since 1991 and
are crucial both in the formulations and proofs of the LCA groups and Fourier transform
approximation theorems. The proofs of GC1 and GC2 combine some methods of NSA
with Fourier-analytic methods of additive combinatorics, stemming from the paper [GR]
by Green and Ruzsa and the book [TV] by Tao and Vu. The proof of GC3 relies on a
fairly general nonstandard version of the Smoothness-and-Decay Principle.
Depending on time, we will survey most of the above mentioned constructions and
results.
Jan. 27, 2015
Dima Sinapova :
4 p.m. in SEO 427
Abstract
Stationary reflection is a compactness-type property and it follows from large cardinals. As such it is at odds with principles like square and failure of SCH. We will discuss how much stationary reflection is
consistent with the failure of SCH.
Feb. 3, 2015
Allen Gehret :
4 p.m. in SEO 427
Abstract
The differential-valued field $\mathbb{T}_{\log}$ of logarithmic transseries is conjectured to have good model theoretic properties. As a partial result in this direction, and as a
confidence building measure we prove that at least its \emph{asymptotic couple} has a good model theory. The value group $\Gamma_{\log}$ of $\mathbb{T}_{\log}$ can be given the additional
structure of a map $\psi:\Gamma\to\Gamma$ which is induced by the derivation on $\mathbb{T}_{\log}$. The structure $(\Gamma_{\log},\psi)$ is the asymptotic couple of the field of logarithmic
transseries (in the sense of Rosenlicht). In this talk we will discuss the good model-theoretic properties of $(\Gamma_{\log},\psi)$, including a quantifier-elimination result in an appropriate
first-order language, definable functions on a certain discrete set, a stable embedding result, and NIP (the Non-Independence Property). All results in this talk (besides NIP)
are in <a href="http://arxiv.org/abs/1405.1012">http://arxiv.org/abs/1405.1012</a>.
Feb. 24, 2015
Christian Rosendal :
4 p.m. in SEO 427
Abstract
We discuss the problem of deciding when a metrisable topological
group G has a canonically defined geometry in an identity neighbourhood. This naturally
leads to the concept of minimal metrics on G, that we characterise in terms
of a linear growth condition on powers of group elements.
In turn, minimal metrics connect with Hilbert’s fifth problem for completely
metrisable groups and we show, assuming that the set of squares is sufficiently
rich, that every element of some identity neighbourhood belongs to a 1-parameter subgroup.
March 3, 2015
Isaac Goldbring :
4 p.m. in SEO 427
Abstract
An operator space is a norm closed linear subspace of the Banach space B(H) of bounded linear operators on a Hilbert space. For reasons that will be explained in this talk, operator spaces are the noncommutative analogs of Banach spaces. A fundamental result of Junge and Pisier shows that there are many more operator spaces than there are Banach spaces in a way to be made precise in the talk. I will explain the model-theoretic content of their result. Parts of this talk represent joint work with Martino Lupini and other parts represent joint work with Thomas Sinclair.
March 10, 2015
Florent Baudier :
4 p.m. in SEO 427
Abstract
We introduce the notion of almost Lipschitz embeddability and study the almost Lipschitz embeddability of proper metric
spaces into Banach spaces. We will discuss the relevance of our work in topology and geometric group theory.
We intend to open the seminar with a brief review (which should be accessible to non-specialists) of some interesting aspects of metric embedding theory.
March 17, 2015
Kathryn Mann :
4 p.m. in SEO 636
Abstract
When is an abstract homomorphism between topological groups forced to be continuous? In geometry, there are several examples of this kind of rigidity or "automatic continuity" using extra assumptions on the groups and homomorphisms involved. Richer algebraic structures -- e.g. Banach algebras, Polish groups -- often exhibit automatic continuity with very few strings attached. The group of homeomorphisms of a compact manifold is one such example. In this talk, I'll show that any homomorphism from Homeo(M) to any other separable topological group is necessarily continuous. The proof combines ideas of Rosendal, standard tricks in automatic continuity, and arguments in manifold topology.
March 31, 2015
Gabriel Conant :
3 p.m. in SEO 1227
Abstract
Suppose $M$ is a metric space taking distances in an arbitrary totally ordered commutative monoid $R$. When considered as a discrete first-order structure in a relational language, nonstandard models of the theory of $M$ can no longer be considered as metric spaces over $R$, in a way coherent with the first-order theory. To solve this problem, we construct a monoid extension $R^*$ of $R$, with the property that any model of the theory of $M$ is a metric space over $R^*$ under a "type-definable" metric. In the case that $R$ is countable, and $M$ is the countable Urysohn space over $R$, we use $R^*$ to characterize quantifier elimination for the theory of $M$.
April 14, 2015
Sergei Starchenko :
4 p.m. in SEO 427
Abstract
In the paper "Crossing patterns of semi-algebraic sets" (J. Combin. Theory Ser. A 111, 2005) Alon et al. showed
that families of graphs with the edge relation given by a semialgebraic relation of bounded complexity satisfy a stronger regularity property than arbitrary graphs.
In this talk we show that this can be generalized to families of graphs whose edge relation is uniformly definable in a structure satisfying a certain model theoretic property called distality.
This is a joint work with A. Chernikov.
April 21, 2015
Sherwood Hachtman :
4 p.m. in SEO 427
Abstract
Borel determinacy, though a theorem of ordinary analysis, cannot be proven without some appeal to the higher infinities of set theory: By the dual results of Harvey Friedman and Donald Martin, the strength of $\Sigma^0_{1+\alpha+3}$-determinacy is roughly that of ZF with Power set restricted to $\alpha+1$ iterations on $\omega$.
Refining these results, we eliminate the "roughly", isolating a family of novel reflection principles whose strengths correspond exactly to that of determinacy for these levels. We will describe this work, also mentioning connections with higher-order reverse mathematics, and stronger determinacy principles having the strength of measurable cardinals.
We also discuss some recent work building on that of Philip Welch, giving a new, natural characterization of the strength of $\Sigma^0_3$-determinacy in terms of monotone operators.
April 28, 2015
Nick Ramsey :
4 p.m. in SEO 427
Abstract
A remarkable theorem of Shelah asserts that an unsimple theory is unsimple in one of two ways: either it has the tree property of the first kind or the tree property of the second kind. Artem Chernikov and I recently studied some possible `quantitative refinements' of this theorem, by considering the relations that obtain between several invariants related to the tree property introduced by Shelah in Chapter 3 of Classification Theory. I'll report on this work, as well as some more recent developments involving Shelah's theory of strong colorings.
Aug. 25, 2015
Lou van den Dries :
4 p.m. in SEO 427
Abstract
I will introduce the differential field of transseries, and outline the favorable algebraic and model theoretic results obtained about it in recent years, some just in the last month. There are several
attractive open problems that can probably be approached by existing techniques, and also many that will probably require a new idea. I expect to discuss both kinds. (Joint work with
Matthias Aschenbrenner and Joris van der Hoeven)
Sept. 1, 2015
Isaac Goldbring :
4 p.m. in SEO 427
Abstract
For a set A of natural numbers, let FS(A) denote the set of sums of finitely many distinct elements of A. A set B of natural numbers is said to be an IP set if B contains FS(A) for some infinite set A. A central result in combinatorial number theory is Hindman's theorem, which states that if one finitely colors an IP set, then at least one of the colors is an IP set. The slickest proof of this result uses idempotent ultrafilters. Di Nasso suggested a model-theoretic generalization of idempotent ultrafilters, aptly named idempotent types, and asked in what completions of PA idempotent types exist. In this talk, I will show that Hindman's theorem is actually equivalent to the existence of idempotent types in all countable complete extensions of PA. This has potential philosophical consequences that I will also discuss. This is joint work with Uri Andrews.
Sept. 8, 2015
James Cummings :
4 p.m. in SEO 427
Abstract
A super-Dowker filter is a filter F on a set X such that
1) For every sequence of F-large sets there are x,y distinct with x in A_y and y in A_x
2) For every partition of X into two parts there exist a sequence as in 1) and a cell of the partition such that all pairs as in 1) lie in this cell
Building on work of Balogh and Gruenhage we show the consistency of the existence of a super-Dowker filter.
Sept. 15, 2015
Daniel Palacín :
4 p.m. in SEO 427
Abstract
Given a first-order structure it is natural to ask whether some of its model-theoretic properties are preserved after enriching the structure with additional predicates.
In this talk, after giving some basic definitions on stability theory, I shall discuss whether the additive group of integers, which is a stable structure and whose first-order theory
is well-understood, admits a stable expansion. More precisely, I will present some superstable expansions of infinite rank, and argue why there is no expansion of finite rank.
This is a joint work with Rizos Sklinos.
Sept. 22, 2015
John Baldwin :
4 p.m. in SEO 427
Abstract
At least since Skolem's formulation of his paradox set theory and
model theory have been intertwined. In contrast to Skolem, we
investigate the methodological role of set theory in model theory. We
will address several questions. What is the role of axiomatic set
theory in model theory? How does this depend on whether the object of
study is a logic or a theory? What is the role of combinatorial set
theory in model theory? In particular, what is the role of
indiscernibles in model theory? What is the role of cardinality in
model theory. How do properties of largish cardinals affect countable
countable models? In this talk we will outline some of these issues and discuss some competing
views.
Sept. 29, 2015
Dima Sinapova :
4 p.m. in SEO 427
Abstract
The tree property is a reflection type combinatorial principle.
It holds at $\omega$ (Konig's infinity lemma),
fails at $\omega_1$ (Aronszajn) and can consistently hold at
$\omega_2$ (Mitchell).
More generally, it is a remnant of large cardinals,
but can hold at successor cardinals. A long standing project in set theory
is to try to obtain the tree property at every regular cardinal
greater than $\omega_1$.
We will start by introducing some classical results.
Then I will discuss a recent result that assuming large cardinals,
one can consistently get the tree property at the first and second
successor of a singular strong limit cardinal.
Oct. 6, 2015
Dima Sinapova :
4 p.m. in SEO 427
Abstract
The tree property is a reflection type combinatorial principle.
It holds at $\omega$ (Konig's infinity lemma),
fails at $\omega_1$ (Aronszajn) and can consistently hold at
$\omega_2$ (Mitchell).
More generally, it is a remnant of large cardinals,
but can hold at successor cardinals. A long standing project in set theory
is to try to obtain the tree property at every regular cardinal
greater than $\omega_1$.
We will start by introducing some classical results.
Then I will discuss a recent result that assuming large cardinals,
one can consistently get the tree property at the first and second
successor of a singular strong limit cardinal.
Oct. 8, 2015
Bradd Hart :
4 p.m. in SEO 427
Abstract
In recent years, Boris Zilber has begun a project to establish a duality between coordinate algebras and geometric structures built from Zariski geometries. The most fully worked out part of this project revolves around Heisenberg-Weyl algebras and the canonical commutation relation. In joint work with Martin Bays, we have taken Zilber’s approach and applied it to one of the natural structures of mathematical physics, the space of tempered distributions, in order to show that this space is pseudo-finite and that the canonical commutation relation holds from a model theoretic point of view. Along the way, we make some remarks about cats (Schroedinger and otherwise).
Oct. 13, 2015
Caroline Terry :
4 p.m. in SEO 427
Abstract
Suppose $\mathcal{L}$ is a finite first-order language and for each integer $n$,
suppose $F(n)$ is a set of $\mathcal{L}$-structures with underlying set
$\{1,\ldots, n\}$. We say the family $F=\bigcup_{n\in \mathbb{N}}F(n)$ has
a zero-one law if for every first order sentence $\phi$,
the proportion of elements in $F(n)$ which satisfy $\phi$ goes to zero or one
as $n\rightarrow \infty$. In this talk we give a brief overview of the history
of this topic, then present some new examples of families with zero-one laws.
This is joint work with Dhruv Mubayi.
Oct. 20, 2015
Sebastien Vasey :
4 p.m. in SEO 427
Abstract
Abstract elementary classes (AECs) are an axiomatic framework encompassing classes of models of an $L_{\lambda, \omega}$ sentence, as well as numerous algebraic examples. They were introduced by Saharon Shelah in the mid seventies. One of Shelah's goals was to study generalizations of Morley's categoricity theorem to the infinitary setup. Among several variations, Shelah conjectured the following eventual version: An AEC categorical in a high-enough cardinal is categorical on a tail of cardinals.
In this talk, we will prove the conjecture for universal classes. It is an interesting type of AEC introduced by Shelah in a milestone 1987 paper [Sh:300] (the work was done in 1985). They correspond approximately to classes of models of a universal $L_{\lambda, \omega}$ sentence. The proof of the conjecture proceeds by first observing that any universal class satisfies tameness: a locality property isolated by Grossberg and VanDieren which says that orbital types are determined by their small restrictions. Next, several structural properties are derived from categoricity: the class has amalgamation on a tail and in fact admits a well-behaved forking-like independence relation. Finally, a definition of a unidimensionality-like property (due to Shelah) is shown to follow from categoricity in a single cardinal and imply categoricity on a tail of cardinals. The argument generalizes to tame AECS which have primes over sets of the form $Ma$.
Oct. 27, 2015
Sergei Bezuglyi :
4 p.m. in SEO 427
Abstract
The talk is devoted to Bratteli diagrams, the object that is widely used for constructions of transformation models in various dynamics. A class of graduated infinite graphs, later called Bratteli diagrams, was originally introduced by O. Bratteli for the classification of AF
C*-algebras. During the last two decades, Bratteli diagrams turned out to be a very powerful and productive tool for the study of dynamical systems in ergodic theory, Cantor and Borel dynamics.
We will focus on Cantor dynamical systems and show that a large part of results proved in the context of Cantor minimal dynamical systems remains true for a much wider class of aperiodic homeomorphisms of a Cantor set. Our main results are about invariant measures and their relations with the structure of a Bratteli diagram.
All necessary definitions will be given in the talk.
Nov. 10, 2015
Evgeny Gordon :
4 p.m. in SEO 427
Abstract
In the second half of the last century a new point of view on interrelation
between the continuous and discrete mathematics started to become popular
among applied mathematicians. According to it the continuous mathematics is
an approximation of the discrete one, but not vice versa. The reason of
this popularity is the widespread use of computers in both applied and
theoretical research. However, the formalization of mathematics based on
this point of view meets serious difficulties in the framework of Cantor's Set
Theory, because we need to deal with not well defined collections, like very big
numbers, or numbers far enough of boundaries of computer memory,
that depend on concrete problems or points of views.
Maybe, the difficulties in mathematically rigorous justification of
theoretical physics have the same reason the axiom of least upper bound is
too strong idealization for physics. A new axiomatic system
(<b>NNST</b> — Naive Nonstandard Set Theory) based on ideas of A.
Robinson's Nonstandard Analysis and P. Vopenka's Alternative Set Theory
will be presented in this talk. The idea of approximation of discrete
structures by continuous ones is implemented in this theory as follows.
Continuous structures appear from finite <i>very big finite ones</i> as
factorizations of <i>accessible substructures</i> of these finite structures
by some <i>indiscirnability relations</i>. The properties in italic here are
not well defined ones. We discuss some theorems formulated and proved in
the framework of NNST related to computer simulations of continuous
structures, which have clear intuitive sense, can be monitored in
computer experiments, but whose formulations in the framework of
Cantor's Set Theory are irrelevant, if not to say unreadable.
Nov. 12, 2015
Monica VanDieren :
4 p.m. in SEO 427
Abstract
In the 1970s Saharon Shelah initiated a program of generalizing first order model theoretic results to the setting of abstract elementary classes (AECs). Progress has been slow due in part to the difficulty of working without the compactness theorem. In this talk we will survey some of the recent developments in understanding superstability in AECs. In particular, we will discuss the relationship between symmetry and the uniqueness of limit models, and we will sketch the proof that in suitably superstable AECs the union of an increasing chain of saturated models is saturated.
Nov. 17, 2015
Phillip Wesolek :
4 p.m. in SEO 427
Abstract
(Joint work with Jay Williams) The space of marked groups is a cantor space that parameterizes all countable groups. This space allows for tools from descriptive set theory to be applied to group-theoretic questions. The class of elementary amenable groups is the smallest class that contains the abelian groups and the finite groups and that is closed under group extension, taking subgroups, taking quotients, and taking directed unions. In this talk, we first give a characterization of elementary amenable marked groups in terms of well-founded trees; as a consequence, elementary amenability is equivalent to a chain condition. We then show the set of elementary amenable marked groups is coanalytic and non-Borel. This gives a new, non-constructive proof of a theorem of Grigorchuk: There are amenable non-elementary amenable groups. We conclude by discussing further questions and possible generalizations of the techniques.
Nov. 24, 2015
Sherwood Hachtman :
4 p.m. in SEO 427
Abstract
Infinite time Turing machines, introduced by Hamkins and Kidder, extend the usual notion of Turing computability by allowing the machine to proceed for an arbitrary ordinal number of steps. These machines may either halt or enter a loop at some countable ordinal stage; thus there are several feasible notions of "Turing jump" for infinite time Turing computability. I will discuss a recent result of Philip Welch illustrating an intimate connection between the jump operator identifying those computations which "eventually settle" (loop with fixed output), and $\Sigma^0_3$ determinacy.
Jan. 19, 2016
Isaac Goldbring :
4 p.m. in SEO 427
Abstract
In the early 2000s, Renling Jin used nonstandard analysis to prove the following result: if $A$ and $B$ are subsets of the integers of positive Banach density, then $A+B$ is piecewise syndetic, meaning that there is a natural number $m$ such that $A+B+[-m,m]$ contains arbitrarily long intervals. Jin’s result had subsequently been improved upon in two different ways. First, Beiglbock, Bergelson, and Fish generalized Jin’s result to arbitrary amenable groups. Secondly, in joint work with DiNasso, Jin, Leth, Lupini, and Mahlburg, we proved a ``quantitative version'' of Jin’s result for the integers by giving a lower bound on the density of the set of witnesses to piecewise syndeticity. In this talk, I will prove a common generalization of these two results (again joint with DJLLM) by proving the amenable group version of our quantitative result.
Jan. 26, 2016
Meng-Che Ho :
4 p.m. in SEO 427
Abstract
In this talk, we will consider two complexity notions of computable groups - the syntactic complexity of a computable Scott sentence and the $m$-degree of the index set of a group. Finding the exact complexity of one of them usually involves finding the complexity of the other, but this is not always the case. Knight et al.~determined the complexity of index sets of various structures.
Generalizing methods that was previously used by Knight et al., we give computable Scott sentences for various different groups, including nilpotent groups, polycyclic groups, certain solvable groups, and certain subgroups of $\mathbb{Q}$. In some of these cases, we also show that the sentence we give are optimal. We also give an example showing d-$\Sigma_2\subsetneq\Delta_3$ in the complexity hierarchy of pseudo-Scott sentences, contrasting the result by D. Miller saying d-$\boldsymbol{\Sigma}_2=\boldsymbol{\Delta}_3$ in the complexity hierarchy of Scott sentences, which is related to the boldface Borel hierarchy.
Feb. 2, 2016
Dima Sinapova :
4 p.m. in SEO 427
Abstract
We will show that the tree property can consistently hold at $\aleph_{\omega^2+1}$ and $\aleph_{\omega^2+2}$ simultaneously, where $\aleph_{\omega^2}$ is strong limit. This result is motivated by the long term project in set theory to get the tree property at every regular cardinal greater than $\aleph_1$. This is joint work with Spencer Unger.
Feb. 9, 2016
Sean Cox :
4 p.m. in SEO 427
Abstract
By results of Viale-Weiss, stationarity of the class of "guessing models" is responsible for many of the consequences of the Proper Forcing Axiom (PFA), including the Tree Property at $\omega_2$ and failure of square. I will discuss joint work with Krueger about the stronger notion of "indestructible guessing models". Stationarity of such models captures even more consequences of PFA (e.g. Suslin's Hypothesis, SCH, and a theorem of Todorcevic), but on the other hand doesn't decide the value of the continuum.
Feb. 16, 2016
Aristotelis Panagiotopoulos :
4 p.m. in SEO 427
Abstract
In every dimension $n$, there exists a canonical compact, metrizable space called the $n$-dimensional Menger space. For $n=0$, it is the Cantor space and for $n=\infty$, it is the Hilbert cube. On the first part of the talk I will illustrate how basic notions of classical descriptive set theory naturally generalize into higher homotopical dimensions. In the second part of the talk I show how projective Fraisse machinery can be employed in the study of the Menger compacta.
This is a joint work with Slawomir Solecki.
Feb. 23, 2016
Christian Rosendal :
4 p.m. in SEO 427
Abstract
We shall present a characterisation and various results concerning the class of Polish groups of bounded geometry, i.e., groups coarsely equivalent to proper metric spaces. As the extent of this class is still somewhat unclear, several fundamental questions remain open, in particular as related to earlier work of Gromov and Eskin, Fisher and Whyte.
March 1, 2016
Philipp Hieronymi :
4 p.m. in SEO 427
Abstract
Let $R$ denote the real ordered field. Our focus here is on expansions of $R$ by Cantor sets. For our purposes, a Cantor set is a non-empty, compact subset of the real line that has neither interior nor isolated points. We consider the following question due to Friedman, Kurdyka, Miller and Speissegger: is there a Cantor set $K$ and a natural number $N$ such that every set definable in $(R,K)$ is $\Sigma_N^1$? I will answer this question positively. In addition to using techniques from model theory, o-minimality and descriptive set theory and previous work of Friedman et al., the work presented in this talk depends crucially on well known results about the monadic second order theory of one successor due to Buechi, Landweber and McNaughton.
March 8, 2016
Henry Towsner :
4 p.m. in SEO 427
Abstract
The Aldous-Hoover Theorem gives a characterization of those random processes which generate "exchangeable" first-order structures. A random first-order structure on the natural numbers is exchangeable if, after any permutation of the natural numbers, it has the same distribution. Although combinatorial proofs now exist, the original proof was model-theoretic: one views as exchangeable process as one given by sampling countably many points from an ultraproduct according to its Loeb measure.
For some purposes, full exchangeability is too strong. We investigate "relative exchangeability", where we only require that the distribution be preserved by automorphisms of a fixed first-order structure M. Depending on the amalgamation properties of the finite substructures of M, we obtain various generalizations of the Aldous-Hoover Theorem to this setting.
March 15, 2016
Jan Reimann :
4 p.m. in SEO 427
Abstract
Diophantine approximation studies how well real numbers can be approximated in terms of rational numbers (or more generally, algebraic numbers). One measure of approximability is the irrationality exponent -- the supremum of all numbers $r>0$ such that there exist infinite many rational numbers $p/q$ with $|x - p/q| < 1/{q^r}$.
Almost every number (with respect to Lebesgue measure) has irrationality exponent 2. In this talk, we present a new result that strengthens and effectivizes a classical theorem due to Jarnik and Besicovitch regarding the Hausdorff dimension of sets of reals with a fixed irrationality exponent.
March 17, 2016
Siddharth Bhaskar :
4 p.m. in SEO 427
Abstract
There are various ways to define the notion of a computable function or predicate over an arbitrary first-order structure. For example, we may specify some model of computation, such as a register machine or system of recursive programs, or we might encode elements of the structure by natural numbers and say that a function is computable when it lifts to a recursive function over N. Given a notion of computability, we can then ask how similar the recursion theory of our structure is to classical recursion theory over N. For example: do we have recursive pairing and unpairing functions? Do we have a Godel numbering and a corresponding evaluation function? Can we compute every recursive function with a tail recursion?
We will survey the general picture and focus on the case most dissimilar from the classical one: namely, functions computable by recursive programs over locally finite structures, where the answers to the above questions tend to be negative. We shall show that for several such structures, computability corresponds to classical complexity classes. We shall finish with a case study of computability over the algebraic closure of a finite field of characteristic p, as well as some open questions.
March 29, 2016
Miguel Angel Mota :
4 p.m. in SEO 427
Abstract
In the last years there has been a second boom of the technique of forcing with side conditions (see for instance the recent works of Asper\'{o}-Mota, Krueger and Neeman describing three different perspectives of this technique). The first boom took place in the 1980s when Todorcevic discovered a method of forcing in which elementary substructures are included in the conditions of a forcing poset to ensure that the forcing poset preserves cardinals. More than twenty years later, Friedman and Mitchell independently took the first step in generalizing the method from adding small (of size at most the first uncountable cardinal) generic objects to adding larger objects by defining forcing posets with finite conditions for adding a club subset on the second uncountable cardinal. However, neither of these results show how to force (with side conditions together with another finite set of objects) the existence of such a large object together with the continuum being small. In this talk we will discuss new results in this area. This is joint work with John Krueger.
April 7, 2016
Andrés Caicedo :
4 p.m. in SEO 427
Abstract
This is joint work with Jacob Hilton. We considered the topological version of the partition calculus in the setting of countable ordinals: Given ordinals $\alpha,\beta_0,\beta_1$, we say that $\alpha\to_{top}(\beta_0,\beta_1)^2$ iff for any 2-coloring of the edges of the complete graph on $\alpha$ vertices, for some color $i$, there is a complete monochromatic graph in color $i$ whose set of vertices is homeomorphic to $\beta_i$. If we insist that $\alpha,\beta_0$ are countable and that $\beta_0>\omega$, then $\beta_1$ must be finite (even without the topological requirement). On the other hand, we have proved that for any countable $\beta_0$ and finite $\beta_1$, we can find a countable $\alpha$ such that $\alpha\to_{top}(\beta_0,\beta_1)^2$. This is a topological version of the Erdős-Milner theorem. Our arguments provide explicit bounds. I will discuss some of these results.
April 12, 2016
Jorge Cely :
4 p.m. in SEO 427
Abstract
This talk will be divided in two parts. In the first part, I will introduce Cluckers-Loeser theory of motivic integration, stressing the role of the model theory of valued fields in the Denef-Pas language. I will talk about transfer principles (in the style of Ax-Kochen).
Then, in the second part, I will discuss the definability (in the Denef-Pas language) of some objects in harmonic analysis of p-adic groups. I will not have time to talk about the fundamental lemma with some detail, nevertheless, I will mention it and roughly I will explain how to put it in the framework of motivic integration.
April 19, 2016
Maxwell Levine :
4 p.m. in SEO 427
Abstract
The combinatorial properties of large cardinals tend to clash with those satisfied by Gödel's constructible universe, especially the square property (denoted $\square_\kappa$) isolated by Jensen in the seventies. Strong cardinal axioms refute the existence of square, but it is possible with some fine-tuning to produce models that exhibit some large cardinal properties together with weakenings of square. In this talk we will exhibit some results along these lines and will outline the techniques used to produce them.
April 21, 2016
T. E. S. Raghavan :
4 p.m. in SEO 636
Abstract
While every win-lose two person extensive game with finitely many moves and finitely many actions in each move and with perfect information admits an optimal winning strategy, it can fail to be true once the number of moves is countable even if the action sets for the players are finite in each move. Gale and Stewart introduced this class of games and showed that open sets or closed sets defined on the terminal vertices of the infinite game tree as winning sets admit winning strategies and hence those games are determined. Blackwell showed that $G_\delta$ set as winning set on the terminal vertices are also determined. Martin proved the remarkable theorem that if the winning set is a Borel subset of the terminal vertices, then also such games are determined.
While zero sum two person stochastic games with finitely many states and actions admit stationary optimal strategies for discounted payoffs, the existence of value with Cesaro payoff is possible only in the space of behavioral strategies. At each move players may have to peg on the entire history so far to make their randomized action choices. The seminal theorem of games is made quite simple and transparent by an application of the above theorem of Martin on perfect information Gale Stewart games. It simply bypasses many complicated constructions and technical estimates that view solution to the discounted Shapley value equation as an elementary sentence in the space of the ordered field of Laurentz series in fractional powers of the discount factor, namely the real closed field of Puiseux series and hence relies on many tools from real algebraic geometry. The talk will present this proof due to Ahok Maitra and Sudderth. Their theorem is applicable to much more general classes of payoffs besides Cesaro payoffs. This is a fruitful interplay between mathematical logic and game theory.
April 26, 2016
Artem Chernikov :
4 p.m. in SEO 427
Abstract
We discuss combinatorial properties of generically stable Keisler measures in NIP theories, concentrating on the improved regularity lemmas for definable (hyper-)graphs in this context.
We give a model-theoretic version of the Lovasz-Szegedy result, generalizing the distal and the stable cases.
This questions are closely related to the existence of "large" homogeneous subsets (in various senses), and we give some partial results and counterexamples, in the o-minimal case and in the p-adics.
Joint work with Sergei Starchenko.
April 27, 2016
John Baldwin :
10 a.m. in SEO 636
Abstract
Emulating Wigner's famous essay we attempt to delineate the
characteristics of model theory that account for its impact across
mathematics. The formalization of specific areas of
mathematics is the basic theme; this allows axiomatizations that
respect the methodologies of each area. Secondly, classification
theory allows the recognition of common methodologies in widely
distinct areas. Thus two large groups of tame areas are
identified: stable (and refinements) and o-minimal. Bourbaki's `great
mother structures': groups, order, topology' are put in perspective
and a 4th mother structure, geometry, takes its place in establishing
dimension as the key to tameness. This organizational survey will be
fleshed out by more specific considerations of interactions with
number theory, identifying specific unifying model theoretic
techniques. Examples include Wilkie-Pila on the Andre-Oort
conjectures and Hrushovski on the function field Mordell-Lang.
Aug. 30, 2016
James Freitag :
4 p.m. in SEO 427
Abstract
The talk will be partly an introduction to differential algebraic geometry, in the sense of Kolchin. We will define various model theoretic ranks on the varieties this geometry, which are special cases of model theoretic notions on general definable sets. Calculating these ranks is not usually very easy, and we will discuss some recent results and open problems.
Sept. 13, 2016
Kostyantyn Slutskyy :
4 p.m. in SEO 427
Abstract
A time change equivalence between free Borel $\mathbb{R}^n$-flows is defined to be an orbit equivalence which is also a homeomorphism when restricted onto any orbit. This notion appeared first in the set up of ergodic theory, when flows and orbit equivalence maps are required to preserve given probability measures, and all constructions are defined up to null sets. It is known that in this case there are continuumly many pairwise non equivalent ergodic $\mathbb{R}$-flows, but, surprisingly, Rudolph showed that any two free ergodic $\mathbb{R}^n$-flows, $n \ge 2$, are time change equivalent.
In the context of Borel dynamics, Miller and Rosendal proved that all $\mathbb{R}$-flows are time change equivalent. They also posed a question of whether Rudolph's theorem is true in the Borel framework. We shall discuss a partial result in this direction, which shows that all free $\mathbb{R}^n$-flows are time change equivalent up to a compressible set.
Sept. 20, 2016
Noah Schweber :
4 p.m. in SEO 427
Abstract
We will discuss a number of set theoretic issues in computable structure theory, especially around the extension of computability-theoretic ideas to uncountable structures. This work is variously solo and joint with Uri Andrews, Greg Igusa, Julia Knight, Joe Miller, Antonio Montalban, and Mariya Soskova.
Sept. 27, 2016
Dima Sinapova :
4 p.m. in SEO 427
Abstract
A remarkable theorem of Shelah states that if $\kappa$ is a singular strong limit cardinal of uncountable cofinality, then there is a subset $x$ of $\kappa$, such that $HOD_x$ contains the powerset of $\kappa$. We show that in general this is not the case for countable cofinality. Using a version of diagonal supercompact extender Prikry forcing, we construct a generic extension in which there is a singular cardinal $\kappa$ with countable cofinality, such that $\kappa^+$ is supercompact in $HOD_x$ for all $x\subset\kappa$. This result was obtained during a SQuaRE meeting at AIM and is joint with Cummings, Friedman, Magidor, and Rinot.
Oct. 18, 2016
Erik Walsberg :
4 p.m. in SEO 427
Abstract
We discuss first order expansions of $(\mathbb{R},<,+)$. In particular we discuss the classification of such structures according to the topology and geometry of their definable sets.
Oct. 25, 2016
John Baldwin :
4 p.m. in SEO 427
Abstract
We construct a complete $L_{\omega_1,\omega}$-sentence $\phi$ such that $(\textbf{R},\subseteq)$ is an abstract elementary class
with a proper class of models.
<b>Theorem.</b> There is a maximal model $M \in \textbf{R}$ of cardinality
$\lambda$ if there is no measurable cardinal $\rho$ with $\rho \leq
\lambda$, $\lambda = \lambda^{< \lambda}$, and there is an $S
\subseteq S^{\lambda}_{\aleph_0}$, that is stationary non-reflecting,
and $\diamond_S$ holds.
Thus in the absence of a measurable, $\phi$ has arbitrarily large maximal models. But in the presence of measurables there are maximal models
cofinally in the first measurable and never again.
I hope to say something about the removal of the set-theoretic hypotheses.
Oct. 28, 2016
Will Boney :
2 p.m. in SEO 427
Abstract
We discuss some characterizations of large cardinals using model theory, especially around compactness in $\mathbb{L}_{\kappa, \kappa}$.
Nov. 1, 2016
Alex Kruckman :
4 p.m. in SEO 636
Abstract
The existence of a robust categorical dual to first-order logic is hinted at in (at least) four independent bodies of work: (1) The cologic of profinite groups (e.g. Galois groups), which plays an important role in the model theory of PAC fields [Cherlin - van den Dries - Macintyre, Chatzidakis]. (2) Projective Fraïssé theory [Solecki & coauthors, Panagiotopolous]. (3) Universal coalgebra and coalgebraic logic [Rutten, Kurz - Rosicky, Moss, others]. (4) Ultracoproducts and coelementary classes of compact Hausdorff spaces [Bankston]. In this talk, I will propose a natural syntax and semantics for such a dual "cologic", in which "coformulas" express properties of partitions of "costructures", dually to the way in which formulas express properties of tuples from structures. I will show how the basic theorems and constructions of first-order logic (completeness, compactness, ultraproducts, Henkin constructions, Löwenheim-Skolem, etc.) can be dualized, and I will discuss some possible extensions of the framework.
Nov. 15, 2016
Garrett Ervin :
4 p.m. in SEO 427
Abstract
Does there exist a linear order that isomorphic to its lexicographically ordered cube but not to its square? Sierpinski posed this problem in his 1958 textbook on set theory <i>Cardinal and Ordinal Numbers</i>. The corresponding question has been answered positively for many other classes of structures, including groups, rings, topological spaces, Boolean algebras, and graphs. However, the answer to Sierpinski’s question turns out to be negative: every linear order isomorphic to its cube is already isomorphic to its square. In this talk, we outline a proof of this result, and connect it with several other problems about linear orders.
Nov. 22, 2016
Victoria Noquez :
4 p.m. in SEO 427
Abstract
In recent years, some progress has been made towards understanding uncountable categoricity in the continuous setting, particularly in the context of classes of Banach spaces. Currently, it is unknown if the Baldwin-Lachlan characterization of uncountable categoricity holds in continuous logic. Namely, is it the case a continuous theory T is kappa-categorical for some uncountable cardinal kappa if and only if T is omega-stable and has no Vaughtian pairs?
In order to address this question, we provide the necessary continuous characterization of Vaughtian pairs, and in the process, prove Vaught's two-cardinal theorem, as well as a partial converse of the theorem in the continuous setting. This allows us to prove the forward direction of the Baldwin-Lachlan characterization.
Trying to prove the reverse direction leads us to an attempt to characterize strong minimality in continuous logic. We propose a notion of strong minimality, and show that it has many of the properties of its classical analogue. Unfortunately, we find that it fails to provide the required machinery, and in fact, conjecture that the reverse direction of the Baldwin-Lachlan characterization is false for continuous logic.
Nov. 29, 2016
Siddharth Bhaskar :
4 p.m. in SEO 427
Abstract
We introduce a novel measure of complexity of set systems called thicket
density which can be seen as an analogue to both VC density and Cantor-
Bendixson rank of topological spaces. The analogue to VC theory is quite
exact: we have "thicket" versions of dimension, shatter function, and density.
In this case, infinite dimensionality captures exactly those set systems which
satisfy the order property, as opposed to the independence property. These
quantities satisfy several identities (including the Sauer-Shelah dichotomy) that
hold verbatim in the VC case. Curiously, the proofs of these identities seem
quite different, and we cannot infer the "thicket" versions from the VC versions
and vice versa.
On the other hand, thicket density can be seen as a finitary version of Cantor-Bendixson rank; in particular, the polynomial/exponential growth dichotomy
from "thicket Sauer-Shelah" corresponds to the countable/continuum cardinality dichotomy in the perfect set theorem. (This justifies the name "Finitary
Stability.") We even believe there is a notion of "derivative of a set system"
that corresponds to the Cantor derivative and induces something like the actual
discrete deriative on shatter functions.
Finitary stability has at least one application in pure model theory, namely
a short proof of the Erdös-Hajnal property for stable graphs, which we might go
over, time permitting. Curiously enough, I discovered it not through model theory, but implicitly hidden in a 1989 paper---about something totally different---by
Polish computer scientist Jerzy Tiuryn.
This talk should be widely accessible, being relatively close to first principles.
It describes work that is very current, and I would like to present one proof (of
"thicket Sauer-Shelah").
Dec. 15, 2016
Mike Lieberman :
4 p.m. in SEO 427
Abstract
Bootstrapping structural properties, via accessible images
We discuss recent joint work with J. Rosicky, [LR], involving new applications of an as-yet-underappreciated tool for the analysis of categories of structures arising in abstract model theory, namely that, under the assumption of sufficiently strongly compact cardinals, the (powerful) image of any accessible functor is accessible ([MP], refined in [BrR]). Although some of these applications are technical (tameness of abstract elementary classes, i.e. AECs, in [LR16]; strong metric tameness of metric AECs in [LR]), we focus on two very simple ones: the amalgamation and joint embedding properties. The basic insight is that each property amounts to a question of the following form: given a diagram of shape A in category of structures K, can it be completed to a diagram in K of shape A'?
We can then rephrase this problem in terms of the forgetful functor U:K^{A'}--->K^A, whose (powerful) image consists precisely of the completable diagrams. As U is accessible, the theorem of [BrR] implies that this image is k-accessible for sufficiently strongly compact k, from which it follows that completability of diagrams involving objects of size up to k implies the completability of diagrams of objects of arbitrary size. That is, this is precisely what one needs to bound the Hanf numbers for amalgamation and joint embedding. This generalizes the results of [BaBo] from AECs to general accessible categories, thereby encompassing e.g. metric AECs, mu-AECs, and so on, with the added benefit of being almost purely visual, replacing delicate syntactic manipulations with simple questions about diagrams in well-behaved categories.
[BaBo] Baldwin, John, and Will Boney. Hanf numbers and presentation theorems in AECs. To appear in Beyond First Order Model Theory, CRC Press.
[BrR] Brooke-Taylor, Andrew, and Jiri Rosicky. Accessible images revisited. Submitted, arXiv:1506.01986.
[LR] Lieberman, Michael and Jiri Rosicky. Bootstrapping structural properties, via accessible images. Submitted, arXiv: 1610.07816.
[LR16] -------------------. Classification theory for accessible categories. Journal of Symbolic Logic, 81(1): 3048-3066 (2016).
[MP] Makkai, Michael and Robert Par\' e. Accessible Categories: The Foundations of Categorical Model Theory. AMS 1989.
Jan. 17, 2017
Chris Shaw :
4 p.m. in SEO 427
Abstract
In this talk we present an overview of progress on the question of which weakly o-minimal structures have definable Skolem functions. Given an o-minimal structure ${\mathcal M}$ expanding a group and a convex predicate $U$ interpreting a nonvaluational convex set, the structure $({\mathcal M}, U)$ fails to have definable Skolem functions, even after naming constants. More generally, any structure which is elementarily equivalent to a reduct of a model formed in this fashion will also fail to have definable Skolem functions. As a partial converse, when $({\mathcal M},U)$ is valuational, modulo adding constants, the expanded structure in fact does have definable Skolem functions. Time permitting, we will give some insight into algorithms that may be used to calculate these functions where they are present.
Jan. 24, 2017
Santiago Camacho :
4 p.m. in SEO 427
Abstract
The theory of analyzable functions is based on the idea that certain classes of (germs of) functions equipped with a valuation can be understood by embedding them into fields of generalized series. One such clear example is the case of Analytic functions and Taylor expansions. We are interested in the cases in which this embedding can be obtained in a truncation closed way. That is embeddings in which all truncations of a series in the image of the embedding belong to the image as well. We will recall previously known results for certain classes of valued fields, and when can a truncation closed embedding be extended to specific ring extensions. We will then present results in the cases of differential ring and differential field extensions. We will conclude with an application in the field of Logarithmic-Exponential Transseries.
Jan. 31, 2017
Maxwell Levine :
4 p.m. in SEO 427
Abstract
Current research in singular cardinal combinatorics emphasizes the tension between the reflection properties entailed by large cardinals and the combinatorial properties of Gödel's constructible universe, notably the square principle that was isolated by Jensen. In order to analyze this tension we construct models that exhibit some reflection properties while still satisfying weakenings of the square property. This presents a technical hurdle because the standard forcing techniques for adding square sequences also add non-reflecting stationary sets. In this talk we will look at why this happens.
Feb. 7, 2017
John Baldwin :
4 p.m. in SEO 427
Abstract
We will discuss the substantial mathematical distinction between the proofs of Godel, Herbrand,
and Henkin in what is usually seen as the same theorem. Then, we'll survey later extensions by Beth, Hintikka,
Smullyan, Makkai leading to the plethora of theorems in logics extending first order finishing our story with
the (new?) proof of completeness for continuous logic by Ben Yaacov and Petersen.
Feb. 14, 2017
James Freitag :
4 p.m. in SEO 427
Abstract
Various finiteness statements across a range of mathematical areas are related to the theory of quasi-orders. These statements are often not recognized as originally coming from finiteness results in the theory of well-quasi-orders, and as a result, complicated combinatorial arguments for the results are often originally developed ad hoc in the various settings. In this talk, we will explain how various results in differential algebra can be proved by using the theory of better quasi-orders.
Better quasi-orders are a natural strengthening of the notion of well-quasi-orders, and the class of bqo's is closed under a number of operations which do not preserve the class of wqo's. Among wqo's, all of those found "in nature" have proven to be bqo's. This lecture will attempt to give the audience the necessary tools to decide if bqo theory can be employed to attack a given finiteness problem.
Feb. 21, 2017
Monroe Eskew :
4 p.m. in SEO 427
Abstract
We discuss our result that it is consistent for the nonstationary ideal to be locally saturated at every regular cardinal. This uses a new Prikry-type forcing, which we hope could be useful for other problems.
Feb. 28, 2017
Allen Gehret :
4 p.m. in SEO 427
Abstract
$H$-fields are ordered differential fields which serve as an abstract generalization of both Hardy fields (ordered differential fields of germs of real-valued functions at $+\infty$) and transseries (ordered valued differential fields such as $\mathbb{T}$ and $\mathbb{T}_{\log}$). A \emph{Liouville closure} of an $H$-field $K$ is a minimal real-closed $H$-field extension of $K$ that is closed under integration and exponential integration. In 2002, Lou van den Dries and Matthias Aschenbrenner proved that every $H$-field $K$ has exactly one, or exactly two, Liouville closures, up to isomorphism over $K$. Recently (in arxiv.org/abs/1608.00997), I was able to determine the precise dividing line of this dichotomy. It involves a technical property of $H$-fields called $\lambda$-freeness. In this talk, I will review the 2002 result of van den Dries and Aschenbrenner and discuss my recent contribution.
March 7, 2017
James Cummings :
4 p.m. in SEO 427
Abstract
We present some applications of iterated forcing with non-stationary support to problems about tall cardinals (joint work with Arthur Apter)
March 14, 2017
Filippo Calderoni :
4 p.m. in SEO 427
Abstract
We will discuss the problem of determining the Borel complexity of the analytic quasi-order of embeddability between torsion-free abelian groups in both the settings of the classical and generalized Descriptive Set Theory. Then we will see how we can slightly modify a categorical construction by Przezdziecki to prove that, for every uncountable $\kappa$ such that $\kappa^{<\kappa}=\kappa$, the embeddability between $\kappa$-sized torsion-free abelian groups is as complicated as possible among the analytic quasi-orders defined on the generalized Baire space on $\kappa$.
March 28, 2017
Gabriel Conant :
4 p.m. in SEO 427
Abstract
Stability and sparsity in sets of natural numbers
The additive group of integers is a well-studied example of a stable group, whose definable sets can be easily and explicitly described. However, until recently, very little has been known about stable expansions of this group. In this talk, we examine the relationship between model-theoretic stability of expansions of the form (Z,+,0,A), where A is a subset of the natural numbers, and the number theoretic behavior of A with respect to sumsets, asymptotic density, and arithmetic progressions.
April 4, 2017
Nadja Hempel :
4 p.m. in SEO 636
Abstract
Given a so called nice graph (no triangles, no squares, for any choice of two distinct vertices there is a third vertex which is connected to one and not the other), Mekler considered the 2-nilpotent subgroup generated by the vertices of the graph in which two elements given by vertices commute if and only if there is an edge between them. These groups form an interesting collection of examples from a model theoretic point of view. It was shown that such a group is stable if and only if the corresponding graph is stable and Baudisch generalized this fact to the simple theory context. In a joint work with Chernikov, we were able to verify this result for NIP and even n-dependent theories. This leads to the existence of groups which are (n+1)-dependent but not n-dependent, providing the first algebraic objects witnessing the strictness of these hierarchy (work in progress).
April 11, 2017
Anton Bernshteyn :
4 p.m. in SEO 427
Abstract
The Lovász Local Lemma (the LLL for short) is an immensely useful tool in probabilistic combinatorics, introduced by Erd\H{o}s and Lov\'asz in the mid-1970s. Roughly speaking, it says that if certain ``bad'' events are reasonably unlikely and mostly independent from each other, then there is a positive (although typically very small) probability of avoiding all of them at once. The LLL has found a wealth of applications, especially in the study of graph colorings. Following a recent breakthrough of Moser and Tardos, who developed an algorithmic approach to the LLL, a number of effective versions of the LLL have been established. In this talk, I will review the LLL and some of its classical applications, and then talk about recently developed measurable versions of the LLL and their consequences in Borel combinatorics and ergodic theory.
April 25, 2017
Dave Marker :
4 p.m. in SEO 427
Abstract
In it's natural formulation Schanuel's Conjecture is a $\Pi^1_1$-sentence. We show there is an equivalent $\Pi^0_3$-sentence.
The key to the proof is work on J. Kirby on natural pregeometries in exponential fields. Most of the talk will be devoted to
explaining Kirby's work.
Sept. 5, 2017
Chieu Minh Tran :
4 p.m. in SEO 427
Abstract
We study the model theory of the structure $(\mathbb{F}; <)$ where $\mathbb{F}$ is the algebraic closure of the field of $p$ elements and $<$ is a cyclic ordering on $\mathbb{F}^\times$ induced by an injective group homomorphism $\chi: \mathbb{F}^\times \to \mathbb{C}^\times$. Various model-theoretic properties of the structure turn out to be consequences of number-theoretic behaviors of the character map $\chi$. The results obtained loosely answer a question by van den Dries, Hrushovski, and Kowalski and form parts of a program to investigate the model-theoretic properties of structures where there is a presence of randomness.
Sept. 12, 2017
Jin Du :
4 p.m. in SEO 427
Abstract
Gitik and Rinot proved assuming the existence of a supercompact that it is consistent to have a strong limit cardinal $\kappa$ such that $2^\kappa=\kappa^+$, there is a very good scale at $\kappa$, and diamond fails along some reflecting stationary subset of $\kappa^+\cap \operatorname{cof}(\omega)$. I will force over Gitik and Rinot's model but with a modification to Gitik-Sharon diagonal Prikry forcing to get this result for $\kappa = \aleph_{\omega^2}$.
Sept. 21, 2017
Ruizhang Jin :
4 p.m. in SEO 427
Abstract
We generalize the well-known fact that the equation $\delta(\mathrm {log}\delta x)=0$ is analyzable in but not internal to the constants. We use the logarithmic derivative as a building block to construct analyzable types with a unique analysis of minimal length (up to interalgebraicity). We also look for criteria for a given definable set such that its pre-image under the logarithmic derivative is analyzable in but not internal to the constants.
Sept. 26, 2017
William Chen :
4 p.m. in SEO 427
Abstract
The stick principle is a weakening of Jensen's diamond that asserts that there is a family of infinite subsets of $omega_1$ so that any uncountable subset of $\omega_1$ has some member of the family as a subset. We will give a forcing construction to separate versions of the stick principle which put a bound on the order-type of the subsets in the family. Many open problems remain about the relationship between different variations of this principle, such as the existence of certain club-guessing sequences or Suslin trees, and we will describe some progress in this direction.
Oct. 3, 2017
Joel Nagloo :
4 p.m. in SEO 636
Abstract
From the work of Freitag and Scanlon, we have that the ODEs satisfied by the Hauptmoduls of arithmetic subgroups of $SL_2(\mathbb{Z})$ are strongly minimal and geometrically trivial. A challenge is to now show that same is true of ODEs satisfied by the Hauptmoduls of all (remaining) Fuchsian triangle groups. The aim of this talk is to both explain why this an interesting/important problem and also to discuss some of the progress made so far.
Oct. 10, 2017
Douglas Ulrich :
4 p.m. in SEO 427
Abstract
Borel Complexity and the Schroder-Bernstein Property
I describe some new techniques for proving non-Borel reducibility results, and give some applications, including: suppose the collection of countable models of a sentence sigma of L_{omega_1 omega} satisfies the Schroder-Bernstein property, that is, if two countable models are bi-embeddable then they are isomorphic. Then, assuming a mild large cardinal, sigma is not Borel complete.
Oct. 24, 2017
Sherwood Hachtman :
4 p.m. in SEO 427
Abstract
The ineffable thin list property, ITP($\kappa$), is a tree property-like principle, introduced by Weiss to capture the combinatorial content of supercompactness. For $\kappa$ inaccessible, ITP($\kappa$) holds if and only if $\kappa$ is supercompact. On the other hand, it is consistent for ITP to hold at accessible cardinals (e.g. $\aleph_2$), and such instances still entail some of the same consequences as supercompactness. For example, in analogy with Solovay's theorem that SCH holds above a supercompact cardinal, Viale has shown that the singular cardinals hypothesis (SCH) holds above $\kappa$ assuming a strengthening of ITP($\kappa$) asserting the existence of stationarily many internally unbounded guessing models.
Does ITP($\kappa$) alone imply the SCH above $\kappa$? We discuss some recent results pointing towards a negative answer. This is joint work with Dima Sinapova.
Nov. 7, 2017
Paul Larson :
4 p.m. in SEO 427
Abstract
We produce a model of ZFA (set theory with atoms) in which the Axiom of Choice holds for pure sets,
but which has no cardinal-preserving outer model of Choice. The construction uses an infinitary sentence (introduced by Hjorth),
having no model of cardinality $\aleph_{2}$, whose unique countable model is highly homogeneous. This is joint work with Saharon Shelah. This answers a question of Eric Hall.
Nov. 14, 2017
Erin Caulfield :
4 p.m. in SEO 427
Abstract
In this talk, I will discuss the progress made towards classifying expansions of the real field by finitely generated subgroups of the complex numbers.
Nov. 21, 2017
John Baldwin :
4 p.m. in SEO 427
Abstract
We attempt to delineate the characteristics of model theory that
account for its impact across mathematics. The {\em formalization}
of {\em specific} areas of mathematics is the basic theme; this
allows axiomatizations that respect the methodologies of each area.
Secondly, classification theory allows the recognition of common
methodologies in widely distinct areas. We distinguish two ways in
which model theory provides tools to `tame' analysis via first order logic:
{\em Axiomatic Analysis} and {\em Definable Analysis}.
In this talk we focus on
Axiomatic Analysis and specifically in the use of the model theory of
differentially closed fields to address century old problems around the
transcendence of solutions of to Painlev{\'e} equations. Thus we give
some context for recent papers of Pillay, Nagloo, and Freitag.
Nov. 28, 2017
Maryanthe Malliaris :
4 p.m. in SEO 427
Abstract
The talk will motivate some recent work
on Ehrenfeucht-Mostowski models (see arxiv: 1709.04899).
Dec. 5, 2017
Silvain Rideau :
4 p.m. in SEO 427
Abstract
A field is said to be pseudo-p-adically closed (ppc) if it is existentially closed in every regular extension to which each of its p-adic valuation extends. Recent work of Montenegro led to a much better understanding of the model theory of bounded ppc fields (for example, we now know that they are NTP2). But one natural question was left open: elimination of imaginaries.
The goal of this talk will be to show how the lack of interaction between the p-adic valuations of a bounded ppc field can be used to classify its imaginaries and show that they can all be described in terms of the imaginaries induced by each valuation. I will also describe a general criterion for elimination of imaginaries, inspired by the proofs of that result in various simple theories, combining quantifier free invariant extensions of types and amalgamation.
Joint with Samaria Montenegro.)
Jan. 23, 2018
James Freitag :
3:30 p.m. in SEO 427
Abstract
The relationship between machine learning and the independence property (in the sense of model theory) is well-known. This seminar is not about that kind of independence. We will give an example of a natural problem in machine learning whose answer does not follow from ZFC.
Feb. 1, 2018
Ioannis Souldatos :
3:30 p.m. in SEO 427
Abstract
During the talk we will briefly survey known theorems about the amalgamation property and the underlying property of
model-existence, and present some recent
developments, focusing on the effects of set-theory on the amalgamation spectrum.
Feb. 6, 2018
Dima Sinapova :
3:30 p.m. in SEO 427
Abstract
Stronger tree properties capture the combinatorial essence of large cardinals. For an inaccessible cardinals $\kappa$, $\kappa$ is strongly (resp. super) compact if and only of $\kappa$ has the strong (resp. super) tree property. A famous theorem of Solovay is that SCH holds above a strongly compact cardinal. This leads to the question of whether the strong or super tree property imply SCH above. One strategy for a positive answer is to use internally unbounded models. We will show the consistency of the super tree property (ITP) holds at the double successor of a singular together with club many non-internally unbounded models, which points to a negative answer. Our construction uses extender based forcing. This is joint work with Sherwood Hachtman.
Feb. 13, 2018
Trevor Wilson :
3:30 p.m. in SEO 427
Abstract
A generic Vopenka cardinal is an inaccessible cardinal kappa such that for every kappa-sequence of structures in
V_kappa in the same first-order language, an elementary embedding between two of the structures exists in some
generic extension of V.
Because the elementary embedding is not required to exist in V, this is a rather weak large cardinal property:
if $0^\sharp$ exists, then every Silver indiscernible is a generic Vopenka cardinal in L.
We show that generic Vopenka cardinals are closely related to a matter in descriptive set theory,
namely the number of Suslin sets of reals in models of ZF without the axiom of choice.
In particular, we show that ZFC + "there is a generic Vopenka cardinal" is equiconsistent with ZF + DC +
"there is no injection from $P(\omega_1)$ to the pointclass of Suslin sets."
Feb. 27, 2018
James Freitag :
1 p.m. in SEO 427
Abstract
This week, we will cover the fundamental theorem of o-minimal structures: every definable set has a finite decomposition into sets called cells, which have very nice properties (for instance very nice topological properties).
Key points we will touch on are: how nice the cells can be chosen, how the decomposition varies in families, and applications of these ideas.
Anush Tserunyan :
3:30 p.m. in SEO 427
Abstract
We prove a pointwise ergodic theorem for quasi-pmp locally countable graphs, which states that the global condition of ergodicity amounts to locally approximating the means of $L^1$-functions via increasing subgraphs with finite connected components. The pmp version of this theorem was first proven by R. Tucker-Drob using probabilistic methods. Our proof is different: it is constructive and applies more generally to quasi-pmp graphs. It involves introducing a graph invariant, a packedness condition for finite Borel subequivalence relations, and an easy method of exploiting nonamenability. The quasi-pmp setting additionally requires a new gadget for analyzing the interplay between the underlying cocycle and the graph.
March 6, 2018
Caroline Terry :
3:30 p.m. in SEO 427
Abstract
A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs. Given a hereditary graph property $\mathcal{H}$, the speed of $\mathcal{H}$ is the function which sends $n$ to the number of distinct elements in $\mathcal{H}$ with underlying set $\{1,\ldots, n\}$. Not just any function can occur as the speed of hereditary graph property. Specifically, there are discrete ``jumps" in the possible speeds. Study of these jumps began with work of Scheinerman and Zito in the 90's, and culminated in a series of papers from the 2000's by Balogh, Bollob\'{a}s, and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized. In contrast to this, many aspects of this problem in the hypergraph setting have remained unknown. In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds. The jumps in the factorial range turned out to have surprising connections to model theory, which we also discuss. This is joint work with Chris Laskowski.
March 13, 2018
Filippo Calderoni :
3:30 p.m. in SEO 427
Abstract
In this talk we show how to use the Ulm theory to prove that the bi-embeddability and isomorphism relations for countable torsion abelian groups are incomparable up to Borel reducibility. This is joint work with Simon Thomas.
March 20, 2018
Travis Nell :
3:30 p.m. in SEO 427
Abstract
Let $\mathcal R=(R;+,<,\ldots)$ be an o-minimal expansion of an ordered group in language $\mathcal L$.
Consider $S\subset R$. In many cases the $\mathcal L \cup \{P\}$-structure $(\mathcal R, S)$,
which interprets $P$ as membership in $S$ can be well understood.
In these cases, one question that can be asked is when "stable" behavior can occur in such a structure.
Distality, introduced by Pierre Simon in 2011, is a notion of a structure being "purely unstable".
I will consider the case where $S$ is a dense $\mathcal L$-elementary substructure of $\mathcal R$.
While these cases are non-distal, we will demonstrate a characterization of the stable behavior in such a structure.
April 10, 2018
Martin Zeman :
3:30 p.m. in SEO 427
Abstract
It is a well-known fact that that if a background certified
extender model has a Woodin cardinal then it may not have an iteration
strategy. The question whether such an iteration strategy exists turns out
to depend on the universe in which such a model is constructed. We give an
example of such a universe (another example was obtained by Woodin some
time ago), and describe an iteration strategy for the background certified
model K^c constructed in this universe. In this situation the model K^c
may contain many Woodin cardinals, but needs to be tame. Generalizations
for the non-tame case are being considered, but have not been fully worked
out. This is a joint work with Grigor Sargsyan.
April 24, 2018
Natasha Dobrinen :
3:30 p.m. in SEO 427
Abstract
It is a central question in the theory of homogeneous relational structures as to which structures have finite big Ramsey degrees. This question, of interest for several decades, has gained recent momentum as it was brought into focus by Kechris, Pestov, and Todorcevic in 2005. An infinite structure S is homogeneous if any isomorphism between two finitely generated substructures of S can be extended to an automorphism of S. A homogeneous structure S is said to have finite big Ramsey degrees if for each finite substructure A of S, there is a number n, depending on A, such that any coloring of the copies of A in S into finitely many colors can be reduced down to no more than n colors on some substructure S’ isomorphic to S. This is interesting not only as a Ramsey property for infinite structures, but also because of its implications for topological dynamics.
Prior to work of the speaker, finite big Ramsey degrees had been proved for a handful of homogeneous structures: the rationals (Devlin 1979) the Rado graph (Sauer 2006), ultrametric spaces (Nguyen Van Thé 2008), and enriched versions of the rationals and related circular directed graphs (Laflamme, Nguyen Van Thé, and Sauer 2010). According to Nguyen Van Thé , "so far, the lack of tools to represent ultrahomogeneous structures is the major obstacle towards a better understanding of their infinite partition properties." We address this obstacle by providing new tools to represent the homogeneous triangle-free graph and developing the necessary Ramsey theory to deduce finite big Ramsey degrees. The methods developed seem robust enough that correct modifications should likely apply to a large class of homogeneous structures omitting some finite substructure.
May 1, 2018
Jindrich Zapletal :
3:30 p.m. in SEO 427
Abstract
There is a very large class of proper forcings associated with analytic hypergraphs on Polish spaces. Their forcing properties to great extent reduce to combinatorics of the hyperedges. The class is also naturally closed under operations such as the countable support iteration and product, resulting in swift proofs of many existing preservation theorems and of many novel ones.
Aug. 28, 2018
John Baldwin :
3:30 p.m. in 427 SEO
Abstract
With Gianluca Paolini, we constructed families of strongly minimal Steiner
$(\infty,2,k)$ systems for every $k \geq 3$. Here we show that the
$2^{\aleph_0}$ Steiner $(2,3)$-systems are coordinatized by strongly
minimal Steiner quasigroups and the Steiner $(2,4)$-systems are
coordinatized by strongly minimal $SQS$-Skeins. Further the Steiner
$(2,4)$-systems admit Steiner quasigroups but it is open whether their
theory is strongly minimal. We exhibit strongly minimal uniform Steiner
triple systems (with respect to the associated graphs $G(a,b)$ (Cameron and
Webb) with varying numbers of finite cycles. This work inaugurates a
program of differentiating the many strongly minimal sets, whose geometries
of algebraically closed sets are (locally isomorphic) to the original
Hrushovski example, but with varying properties in the object language.
Sept. 4, 2018
Caroline Terry :
3:30 p.m. in 427 SEO
Abstract
The arithmetic regularity lemma for $\mathbb{F}_p^n$ (first proved by Green in 2005) states that given $A\subseteq \mathbb{F}_p^n$, there exists $H\leq \mathbb{F}_p^n$ of bounded index such that $A$ is Fourier-uniform with respect to almost all cosets of $H$. In general, the growth of the index of $H$ is required to be of tower type depending on the degree of uniformity, and must also allow for a small number of non-uniform elements. Previously, in joint work with Wolf, we showed that under a natural stability theoretic assumption, the bad bounds and non-uniform elements are not necessary. In this talk, we present results extending these results to stable subsets of arbitrary finite abelian groups. This is joint work with Julia Wolf.
Sept. 11, 2018
Christian Rosendal :
3:30 p.m. in 427 SEO
Abstract
Answering a problem originating in J.P.R. Christensen's seminal work on Haar null sets, we show that a universally measurable homomorphism between Polish groups is automatically continuous.
Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the consistency strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo ZF+DC, the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on Cantor space has finite chromatic number.
Sept. 18, 2018
Mariya Soskova :
3:30 p.m. in 427 SEO
Abstract
The Turing degrees measure the computability-theoretic complexity of elements of $2^\omega$. We can code other mathematical objects as binary sequences and use the Turing degrees to measure their complexity. However, this does not always lead to a coherent measure of complexity; there may not be a ``canonical'' coding. The enumeration degrees, a natural extension of the Turing degrees, work in some circumstances where Turing degrees fail. For example the enumeration degrees can measure the complexity of continuous functions on the unit interval. In fact, we get a proper subclass of the enumeration degrees: the continuous degrees. A larger subclass, the cototal degrees, arises naturally in symbolic dynamics, graph theory and computable structure theory.
Sept. 25, 2018
Greg Cousins :
3:30 p.m. in 427 SEO
Abstract
In this talk, we will discuss how the Lascar group of a first-order theory, $T$, can be recovered as the fundamental group(-oid) of a certain space associated to the category of models, $\operatorname{Mod}(T)$. We will then discuss some examples illustrating how tools from algebraic topology can be used to compute the Lascar group of a theory. Time permitting, we will discuss generalizations to the context of AECs and questions their higher homotopy. No knowledge of homotopy theory will be assumed. This is joint work with Tim Campion and Jinhe Ye.
Oct. 2, 2018
Omer Mermelstein :
3:30 p.m. in 427 SEO
Abstract
The property of "flatness" of a pregeometry (matroid) is best known in model theory as the device with which Hrushovski showed that his example refuting Zilber's conjecture does not interpret an infinite group. Indeed, the pregeometry associated to any hypergraph via Hrushovski's delta-function is flat, and it is known that any finite flat pregeometry (strict gammoid) can be gotten froma hypergraph.
In this talk, we will explain what flatness actually is, show that the interplay between hypergraphs and flat pregeometries runs deeper than the finite case, present some limited results and conjectures based on these.
Oct. 9, 2018
Sebastian Vasey :
3:30 p.m. in 427 SEO
Abstract
The categoricity spectrum of a class of structures is the class of cardinals k such that the class has exactly one model of cardinality k up to isomorphism. Shelah's eventual categoricity conjecture is the statement that the categoricity spectrum of an abstract elementary class (AEC) is either bounded or contains an end segment. Roughly, an AEC is a concrete accessible category with directed colimits satisfying a few extra properties. AECs generalize, for example, classes of models of a first-order theory, classes of models in several infinitary logics, and finitely accessible categories with all morphisms monomorphisms.
Shelah's eventual categoricity conjecture is one of the main test questions in non-elementary model theory, but despite forty years and many more pages of partial approximations it remains open. Until now, it was not even known to be consistent with large cardinals. In this talk, I will present a joint work with Saharon Shelah, a proof that Shelah's eventual categoricity conjecture follows from a large cardinal axiom (a proper class of strongly compact cardinals). The main tool is the method of multidimensional diagrams, introduced by Shelah in the eighties and later rediscovered by Zilber in his study of pseudoexponential fields. The main technical result is that (assuming large cardinals), an AEC with unbounded categoricity spectrum is excellent, i.e. any finite diagram of object can be amalgamated in a way that is, in some category-theoretic sense, unique. The categoricity conjecture then follows from excellence. The method seems to have a strongly category-theoretic nature, and has many other applications that do not need large cardinals. For example, assuming that cardinal exponentiation is injective (a weakening of the GCH), one gets a full understanding of the categoricity spectrum of AECs with the amalgamation property. Assuming a little bit more than the GCH, the eventual categoricity conjecture also holds for any AEC with no maximal models.
Oct. 16, 2018
James Freitag :
3:30 p.m. in 427 SEO
Abstract
In this talk, we will describe the work of Angluin and Dohrn (2017) where a variation on equivalence query learning is analyzed. Specifically, the teacher chooses counterexamples from a fixed (but arbitrary) probability distribution. Angluin and Dohrn show that the number of expected queries for exactly identifying a target concept grows linearly in the $log (n)$ where $n$ is the size of the domain of the concepts. They also show that no better upper bound can be achieved in terms of the VC-dimension of the concept class. Even when the VC-dimension is one, the upper bound can be achieved.
We will show that the number of expected queries grows linearly in the Littlestone dimension. In many set systems, this produces better bounds than those of Angluin and Dohrn, since the Littlestone dimension is bounded from above by $log(n)$, but in general concept classes, $n$ may be arbitrarily large with fixed Littlestone dimension. Many examples are provided by formulas in stable theories (a formula is stable if and only if it has finite Littlestone dimension).
This is joint work with Hunter Chase.
Oct. 30, 2018
Sherwood Hachtman :
3:30 p.m. in 427 SEO
Abstract
Guessing principles assert the existence of elementary submodels of levels of $V$ with a great deal of absoluteness. These characterize large cardinals, but can consistently hold for small cardinals (like $\aleph_2$) as well. Their close relatives, the strong and super tree properties, share this characteristic; but while guessing principles have a number of combinatorial consequences (e.g. for failures of square, approachability, and the SCH), the case with these tree properties is less clear.
We will introduce all of these principles with a minimum of prerequisites, and discuss a number of results concerning the extent to which guessing models and the super tree property can hold near a singular. This is joint work with Dima Sinapova.
Nov. 6, 2018
Dima Sinapova :
3:30 p.m. in 427 SEO
Abstract
There is a natural tension between violating SCH and compactness properties such as square, the tree property and its strengthenings. On the other hand, in order to obtain compactness at many cardinals simultaneously, we need many failures of SCH. A key questions is if a certain compactness property can hold at $\kappa^+$ for a singular $\kappa$ where SCH fails. I will go over some known facts, and then discuss a recent result on ITP and failure of SCH.
Nov. 13, 2018
James Freitag :
3:30 p.m. in 427 SEO
Abstract
We will discuss some recent new techniques to prove functional transcendence results for automorphic functions of discrete groups acting on the upper half plane.
Nov. 20, 2018
Grigor Sargsyan :
3:30 p.m. in 427 SEO
Abstract
Woodin showed that, assuming the existence of a supercompact cardinal and a class of Woodin cardinals, after collapsing a supercompact cardinal to be countable,
the theory of L(Gamma_{uB}) is sealed. Here, Gamma_{uB} is the collection of the universally Baire sets of reals. We say that the theory of L(Gamma_{uB}) is sealed if for any V-generic
g and a V[g]-generic h, there is an elementary embedding j: L(Gamma_uB)^{V[g]}-> L(Gamma_uB)^{V[g*h]}. It has been conjectured by the speaker that sealing has a weak large cardinal strength,
and its weakness is the reason why the core model induction becomes so much more complicated after passing the threshold given by sealing. In a very recent work, the speaker and Trang
showed that sealing is indeed weak, weaker than a Woodin cardinal that is itself a limit of Woodin cardinals. After stating the relevant theorems we will outline why exactly the core model induction becomes
rather difficult after this threshold.
Jan. 15, 2019
Sam Corson :
3:30 p.m. in 427 SEO
Abstract
Some groups are so discrete that abstract homomorphisms from many of the standard continuous topological groups to such a group always have open kernel. It was known classically that any homomorphism from a completely metrizable group to a free group has open kernel. I'll talk about the history of such results and give a plethora of new examples. Joint work with Greg Conner.
Jan. 24, 2019
Carlos Arreche :
3:30 p.m. in 427 SEO
Abstract
Elliptic hypergeometric functions arose roughly 10 years ago as a generalization of classical hypergeometric functions and q-hypergeometric functions. These special functions enjoy remarkable symmetry properties, like their more classical counterparts, and find applications in mathematical physics.
After interpreting one of these symmetries as a linear difference equation over an elliptic curve, we apply the differential Galois theory of difference equations to show that these functions are always differentially transcendental for “generic” values of the parameters. This is joint work with Thomas Dreyfus and Julien Roques.
Feb. 5, 2019
Paul Larson :
3:30 p.m. in 427 SEO
Abstract
We discuss independence results for forms of the Axiom of Choice proved by forcing over the Solovay model.
Feb. 12, 2019
Filippo Calderoni :
3:30 p.m. in 427 SEO
Abstract
In this talk I shall present the status of the ongoing project that aims at computing the universal minimal flow of automorphism groups of all countable torsion abelian groups. Building on work of Kechris-Pestov-Todorcevic, we shall discuss how to compute the universal minimal flow of $Aut(G)$, for some primary group $G$. More precisely, we shall consider the cases when $G$ is a quasi-cyclic group, or the infinite direct sum of $\omega$ copies of a cyclic $p$-group, for any prime $p$. This is joint work with Gianluca Basso.
Feb. 19, 2019
John Baldwin :
3:30 p.m. in 427 SEO
Abstract
Those who heard a version of this talk in early September will be interested to see unintelligible questions replaced by theorems.
With Gianluca Paolini (in preparation), we constructed, using a variant
on the Hrushovski dimension function,
for every $k \geq 3$, $2^\mu$ families of
strongly minimal Steiner $k$-systems. We study the mathematical properties
of these counterexamples to Zilber's trichotomy conjecture rather than thinking of them as merely exotic examples. In particular the long study of finite Steiner systems in reflected in results that depend on the block size $k$.
A quasigroup is a structure with a binary operation such that for each equation $xy=z$ the values of two of the variables determines a unique value for the third.
The
new Steiner $3$-systems are bi-interpretable with strongly
minimal Steiner quasigroups. For $k >3$, we show the pure $k$-Steiner systems have `essentially unary definable closure' and do not interpret a quasigroup. But we show that for $q$ a prime power the Steiner $q$-systems can be interpreted into specific sorts of quasigroups, block algebras.
We extend the notion of an $(a,b)$-cycle graph arising in the study of finite and infinite Stein triple systems (e.g Cameron-Webb) by introducing what we call the $(a,b)$-path graph of a block algebra. We exhibit theories of strongly minimal block algebras where all $(a,b)$-paths are infinite and others in which all are finite only in the prime model. We show how to obtain combinatorial properties (e.g. 2-transitivity) by either varying the basic collection of finite partial Steiner systems or modifying the $\mu$ function which ensures strong minimality.
Feb. 26, 2019
Iian Smythe :
1 p.m. in 427 SEO
Abstract
Given a countable transitive model of set theory and a notion of forcing in it, there is a natural countable Borel equivalence relation on generic objects over the model; two generics are equivalent if they yield the same generic extension. We study generic reals arising from familiar notions of forcing, e.g., Cohen and random forcing, under this equivalence relation and describe their relative complexity using the techniques of invariant descriptive set theory.
March 5, 2019
James Freitag :
3:30 p.m. in 427 SEO
Abstract
In recent joint work with Casale and Nagloo, we proved that the differential equation satisfied by the automorphic function associated with any Fuchsian group is strongly minimal. In this talk, we will give applications of that work around functional transcendence and special points conjectures.
March 19, 2019
Joshua Wiscons :
3:30 p.m. in 427 SEO
Abstract
It became clear through work of Borovik and Cherlin in 2008 that the classification theory for groups of finite Morley rank (fMr) is sufficiently developed that general questions about permutation groups of fMr can be settled even though the classification itself remains incomplete. This is a particularly salient direction since the study of uncountably categorical theories is intertwined with binding groups of fMr (acting on realizations of types).
Borovik and Cherlin posed several motivating problems, many of which are centered around the notion of ``generically $n$-transitive'' actions. In this talk, we will discuss the status and implications of their conjecture that the only transitive and generically $(n+2)$-transitive group of fMr acting on a set of rank $n$ is $\operatorname{PGL}_{n+1}$ acting naturally on projective $n$-space. Among other things, we will highlight a general approach to constructing a projective geometry in this context, and we will also illustrate how this conjecture is intertwined with minimal fMr representations of the finite symmetric groups. The talk will begin with a brief overview of the fMr landscape---knowledge of advanced theory of groups of fMr will be not be required.
April 9, 2019
Dima Sinapova :
3:30 p.m. in 427 SEO
Abstract
It is an old theorem that if a regular cardinal is singularized to have cofinality $\omega$, while preserving cardinals, then $\square_{\kappa, \omega}$ holds in the outer model.
We will show that this does not generalize to uncountable cofinalities.
In particular, we show that after the right kind of preparation, in the Magidor model of
singularizing $\kappa$ to uncountable cofinality all intermediate forms of square at $\kappa$ fail. This is joint work with Maxwell Levine.
April 16, 2019
Noah Schweber :
3:30 p.m. in 427 SEO
Abstract
Over the last several decades, general determinacy principles have been studied extensively in computability theory and reverse mathematics, and a largely-complete analysis has emerged. However, determinacy principles for *topological* games - such as the Banach-Mazur game - are much less well understood. I'll present some initial results about the Banach-Mazur game for subsets of Baire space, including a full analysis of Borel Banach-Mazur determinacy and some comments on *lightface* versions of determinacy principles.
April 30, 2019
Matthew Foreman :
3:30 p.m. in 427 SEO
Abstract
Weakly compact cardinals are equivalent to the statement that every $\kappa$-complete filter on a Boolean algebra $\mathcal{B}$ of size $\kappa$ can be extended to a $\kappa$-complete ultrafilter on $\mathcal{B}$. One can continue this finitely many times. Can it be continued transfinitely?
Fix a cardinal $\kappa$ and consider the following game $\mathcal G_\gamma$ of ordinal length $\gamma$: Player I plays a a sequence of collections $\mathcal S_\alpha\subseteq P(\kappa)$ of size $\kappa$ and player II plays an increasing sequence of $\kappa$-complete ultrafilters $U_\alpha$ on $\bigcup_{\beta\le \alpha}\mathcal S_\beta$. Player II wins if she can continue playing until stage $\gamma$.
Clearly if $\kappa$ is measurable then II wins the game of any length. Welch asked whether the property that ``II has a winning strategy in $\mathcal G_\gamma$" can hold at a non-measurable cardinal.
The main result in this talk is that if II wins $\mathcal G_{\omega_1}$ then there is a precipitous ideal on $\kappa$ whose quotient has a countably closed dense subset. Hence the answer to Welch's question, at least for $\gamma\ge \omega_1$, is no.
In joint work with Magidor, we prove that it is consistent at a non-measurable cardinal for II to have a winning strategy in $\mathcal G_{\omega_1}$, hence the theorem is not vacuous.
Sept. 10, 2019
Nigel Pynn-Coates :
3:30 p.m. in 427 SEO
Abstract
Pre-H-fields are certain ordered valued differential fields introduced by Aschenbrenner, van den Dries, and van der Hoeven in their work on the model theory of transseries. They showed that the model companion of the theory of pre-H-fields is (almost) the theory of transseries. I will discuss the theory of pre-H-fields with gap 0 and describe its model completion.
Sept. 17, 2019
Dima Sinapova :
3:30 p.m. in 427 SEO
Abstract
ITP is a strengthening of the tree property. Just like the tree property characterizes the combinatorial nature of weakly compact cardinals, ITP characterizes it for supercompact cardinals. An old project in set theory is to get these properties at all regular cardinals greater than $\omega_1$. Doing so would require many failures of SCH. We prove that it is consistent to have ITP at $\aleph_{\omega^2+1}$ together with failure of SCH at $\aleph_\omega^2$. This is joint work with J. Cummings, M. Magidor, I. Neeman, S. Unger, and Y. Hayut.
Sept. 24, 2019
James Freitag :
3:30 p.m. in 427 SEO
Abstract
In this talk, I'll mention some recent results and related open problems in model theory which seem within grasp based on these results.
We will talk about problems related to two quite different areas - algebraic differential equations and machine learning.
Oct. 1, 2019
Maxwell Levine :
3:30 p.m. in 427 SEO
Abstract
Singular cardinals yield surprising results in set theory. After Cohen proved that CH is independent of ZFC, Easton proved that on regular cardinals, the continuum function $\kappa \mapsto 2^\kappa$ is constrained only by the facts that $\lambda \le \kappa \Rightarrow 2^\lambda \le 2^\kappa$ and that $\operatorname{cf}(2^\kappa)>\kappa$. In other words, the ZFC constraints on $\kappa \mapsto 2^\kappa$ are fully characterized relative to the class of regular cardinals. In an unexpected turn, Silver proved that GCH cannot fail for the first time at a singular cardinal of uncountable cofinality. More constraints on the arithmetic of singular cardinals were later discovered by Shelah using his PCF theory.
This opens up a more general scheme of questions: Given a property $P$ of a cardinal $\kappa$, what can ZFC prove about the behavior of $P(\kappa)$ across the class of all cardinals? Does $\{P(\kappa):\kappa<\lambda\}$ ever imply $P(\lambda)$?
For this talk, we will present an Easton-style result for stationary reflection. If $S$ is a stationary subset of a cardinal $\kappa$, the reflection principle $SR(S)$ asserts that every stationary subset of $S$ reflects. Assuming the consistency of a supercompact cardinal, we prove that given a fixed $n<\omega$, there are only a few trivial ZFC constraints on $SR(\kappa \cap \operatorname{cof}(\aleph_n))$ (current work in inner model theory suggests that the large cardinal assumption is close to optimal). The successors of singular cardinals present the greatest hurdle for this result, and require a nonstandard approach to PCF theory.
This is joint work with Sy-David Friedman.
Oct. 8, 2019
Kyle Gannon :
3:30 p.m. in 427 SEO
Abstract
In the context of NIP theories, generically stable measures have three main equivalent definitions. In particular, Hrushovski, Pillay and Simon showed a global Keisler measure is generically stable if any one/all of the following hold: (1) the measure is generically stable (otherwise known as FIM), (2) the measure if finitely approximable, and (3) the measure is definable and finitely satisfiable over a small model. The purpose of this talk is to discuss how these different definitions separate if we remove the NIP assumption. This is joint work with Gabriel Conant.
Oct. 9, 2019
James Hanson :
3 p.m. in 427 SEO
Abstract
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. We will also examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets, and the issue of the number of separable models of an inseparably categorical theory.
Oct. 15, 2019
Ruiyuan (Ronnie) Chen :
3:30 p.m. in 427 SEO
Abstract
It is well-known that countable infinitary model theory is closely related to the dynamics of non-Archimedean Polish groups. We extend this correspondence by showing that there is a complete (2-categorical) equivalence between countable $\mathcal{L}_{\omega_1\omega}$-theories and their open non-Archimedean Polish groupoids of countable models. We will also discuss the extension of this correspondence to continuous logic.
Oct. 22, 2019
Daniel Hoffman :
3:30 p.m. in 427 SEO
Abstract
There are several attempts to describe theories by Galois groups, and new notions of Galois group have been defined for this purpose (Shelah Galois group, Kim-Pillay Galois group, Lascar Galois group). My project goes in the other direction: instead of introducing new Galois groups, finding theories which are controlled by the “classical” Galois groups.
In the case of the theory of fields, there is a special class of fields, pseudoalgebraically closed fields (PAC fields). PAC fields were the core of research in field theory in the second half of the 20th century. Why? Because the theory of a PAC field is controlled by its absolute Galois group, so all the machinery from Galois theory can be invoked and used with success; e.g. Nick Ramsey showed that a PAC field is NSOP1 if and only if its absolute Galois group is NSOP1. Therefore it makes sense to develop model-theoretic Galois theory in the case of PAC structures, a generalization of PAC fields. With my co-authors, I obtained recently a generalization of the Elementary Equivalence Theorem for PAC structures: two PAC structures share the same first order theory provided they have isomorphic absolute Galois groups. Also Ramsey’s result was generalized. In my talk, I will summarize the situation and explain connections between some results from my preprints, because combining them together gives us an algorithm for obtaining PAC structures with an absolute Galois group which can be “calculated”, and so there is a prospective way to generate new examples of NSOP1 structures.
Oct. 29, 2019
Roland Walker :
3:30 p.m. in 427 SEO
Abstract
We develop distality rank as a property of first-order theories and give examples for each rank $m$ such that $1\leq m \leq \omega$. For NIP theories, we show that distality rank is invariant under base change. We also define a generalization of type orthogonality called $m$-determinacy and show that theories of distality rank $m$ require certain products to be $m$-determined. Furthermore, for NIP theories, this behavior characterizes distality rank $m$.
Nov. 5, 2019
Adam Clay :
3 p.m. in 427 SEO
Abstract
A group G is called left-orderable if it admits a total ordering of its elements that is invariant under left multiplication. For a fixed group G, the set of all such orderings LO(G) can be topologized so as to become a compact space, in fact it is Polish whenever the group is countable and homeomorphic to a Cantor set whenever it admits no isolated points. This talk will be an introduction to LO(G), its topological properties and their connections to the algebra of the underlying group. I will also discuss a recent question of K. Mann concerning the structure of LO(G), and a strategy for tackling it in certain special cases.
Nov. 12, 2019
Joseph Zielinski :
3:30 p.m. in 427 SEO
Abstract
By results of A.S. Kechris, whenever a locally compact Polish group acts continuously on a Polish space, the orbit equivalence relation of the action is essentially countable—that is, Borel reducible to the orbit equivalence relation of an action of a countable group. It is unknown if this characterizes the locally compact Polish groups.
S. Solecki, after proving an analogous characterization for smooth actions of compact Polish groups, showed this to be true in the case where the group, G, is the additive group of a separable Banach space. The characterization also holds for abelian pro-countable groups, by results of M. Malicki. We discuss recent work on this problem, including an extension of this characterization to some important classes of Polish groups.
This is joint work with A.S. Kechris, M. Malicki, and A. Panagiotopoulos.
Nov. 19, 2019
Chris Miller :
3:30 p.m. in 427 SEO
Abstract
We consider structures on the set of real numbers having the property that connected components of definable sets are definable. All o-minimal structures on the real line (R,<) have the property, as do all expansions of the real field that define the set N of natural numbers. Our main analytic-geometric result is that any such expansion of (R,<,+) by boolean combinations of open sets (of any arities) is either o-minimal or undecidable. We also show that expansions of (R, <, N) by subsets of N^n (n allowed to vary) have the property if and only if all arithmetic sets are definable. (Joint with A. Dolich, A. Savatovsky and A. Thamrongthanyalak.)
Jan. 28, 2020
David Marker :
3 p.m. in 427 SEO
Abstract
It's been known since work of Harrington in the early 1970s that
computable differential fields have computable differential closures.
Recently Calvert, Frolov, Harizanov, Knight, McCoy, Soskova, and
Vatev showed that the countable saturated differentially closed field is computable.
Their proof involves first creating an effective listing of all types and then
using a result of Morley's on existence of computable saturated models.
I will give a significant simplification of the enumeration result and, for completeness,
sketch Morley's priority construction of a saturated model. Pillay has also
given an alternative enumeration argument though ours seems more robust
and generalizes to quantifier free types in ACFA.
Feb. 4, 2020
James Freitag :
3 p.m. in 427 SEO
Abstract
It is a classical result (combining results of Tits and Hall) that there is no sharply 4-transitive action of a group on an infinite set. For algebraic groups acting on (infinite) varieties, there is no 4-transitive group action. Things get more interesting when we loosen the requirements slightly and merely demand that the action has a "large" orbit for a suitable notion of large.
In this talk we will discuss some conjectures in this area and the current prospects for solving them.
We will also talk about the connection between conjectures in this area and some notions from geometric stability theory.
Feb. 11, 2020
Dima Sinapova :
3 p.m. in 427 SEO
Abstract
There is an inherent tension between stationary reflection and the failure of SCH. The former is a compactness type principle that follows from large cardinals. The latter is an instance of incompactness, and usually obtained using Prikry forcing. We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH. Then we discuss the situation at smaller cardinals. This is joint work with Alejandro Poveda and Assaf Rinot.
Feb. 18, 2020
Filippo Calderoni :
3 p.m. in 427 SEO
Abstract
In this work we address a question of Deroin, Navas, and Rivas. We discuss how descriptive set theory can be used to find many examples of countable left-orderable groups such that the quotient space \(\mathrm{LO}(G)/G\) is not standard.
Moreover, we prove that the countable Borel equivalence relation induced from the conjugacy action of \(\mathbb{F}_{2}\) on \(\mathrm{LO}(\mathbb{F}_{2})\) is universal. This is joint work with Adam Clay.
Feb. 25, 2020
Matt Foreman :
3 p.m. in 427 SEO
Abstract
In his seminal 1932 paper von Neumann asked whether it is possible to tell whether time is going forwards or time is going backward from the statistics of a measure preserving system. It was not until 1941 that Anzai produced the first example of a measure preserving system where $T$ is not isomorphic to $T^{-1}$.
In this talk I show that there is a one-to-one, primitive recursive map $F$ that maps Gödel numbers of $\Pi^0_1$-sentences to recursive, measure preserving, invertible, ergodic diffeomorphisms of the 2-torus such that:
\[\phi \mbox{ is true if and only if } F(\phi) \mbox{ is isomorphic to }F(\phi)^{-1}.\]
As corollaries there are non-isomorphic ergodic measure preserving diffeomorphism of the torus:
- $T_{RH}$ such that $T_{RH}\cong T_{RH}^{-1}$ if and only if the Riemann Hypothesis is true.
- $T_{GC}$ such that $T_{GC}\cong T_{GC}^{-1}$ if and only if Goldbach's Conjecture is true.
- $T_{ZFC}$ such that $T_{ZFC}\cong T_{ZFC}^{-1}$ if and only if ZFC is consistent.
- $T$ such that $T\cong T^{-1}$ but this is independent of ''ZFC + there is a supercompact cardinal."
March 3, 2020
Noah Schoem :
3 p.m. in 427 SEO
Abstract
An ideal $I$ on a set is said to be $\lambda$-saturated if no family of $I$-almost pairwise disjoint $I$-positive sets has cardinality $\lambda$. Sufficiently small saturation allows $I$ to generically create a nontrivial elementary embedding $j:V\to M$ in an outer model, serving as a kind of generic large cardinal axiom.
We exhibit a new anti-saturation result. Inspired by a result of Cox and Eskew, we show that it is possible to force to destroy the saturation of ideals at an inaccessible cardinal, while preserving many of their large cardinal-type properties.
March 10, 2020
John Baldwin :
3 p.m. in 427 SEO
Abstract
We describe Shelah’s construction of atomic models in the continuum, as reformulated with Laskowski as a Henkin construction [2]. Then we discuss briefly the connection with the Ackerman-Freer-Patel [1] proof that if $M$ is a countable structure for
a relational language $L$ with trivial definable closure then there is an invariant probability measures on the countable $L$-structures that concentrates on $M$. We explain
while the sufficient conditions for the model in the continuum include those with trivial definable closure, our theorem applies more generally to obtaining a atomic model
in the continuum of the first order theory of a countable atomic extendible structure
admitting a formula-based geometry.
[1] Ackerman, N. and Freer, C. and Patel, R., Invariant measures concentrated
on countable structures, Forum of Mathematics, Sigmas, vol. 4 (2016), no. X, pp. 59.
[2] Baldwin, J. T. and Laskowski, M.C., Henkin Constructions of Models in the
Continuum, Bulletin of Symbolic Logic, vol. 24(2019), no. 1, pp. 1–34.
March 17, 2020
Remi Jaoui :
3 p.m. in 427 SEO
March 31, 2020
Philipp Hieronymi :
3 p.m. in 427 SEO
April 7, 2020
Jenna Zomback :
3 p.m. in 427 SEO
April 14, 2020
Assaf Shani :
3 p.m. in 427 SEO
April 21, 2020
Grigor Sargsyan :
3 p.m. in 427 SEO
April 28, 2020
Paul Larson :
3 p.m. in 427 SEO
Sept. 1, 2020
Shaun Allison :
2 p.m. in Zoom
Abstract
Two dynamical conditions for orbit equivalence relations will be introduced -- the first providing an obstruction to classification by TSI Polish groups, and the second providing an obstruction to classification by non-Archimedean TSI Polish groups. A Polish group is TSI iff it has a compatible two-sided invariant metric. Following work of Hjorth and Drucker, we mimic the Scott analysis of countable structures to understand actions of general TSI Polish groups. Answering a question of Clemens and Coskey, these methods are used to show that the $\mathbb{Z}$-jump of $E_0$ is not classifiable by TSI Polish groups. Time permitting, we will show that if $E$ is Borel-reducible to $=^+$ and also classifiable by a non-Archimedean TSI Polish group, then it is Borel-reducible to $E_\infty^\omega$. Much of this work is joint with Aristotelis Panagiotopoulos.
Sept. 15, 2020
Ralf Schindler :
11 a.m. in Zoom
Abstract
Forcing axioms spell out the dictum that if a statement can be forced, then it is already true.
The $P_\text{max}$ axiom $(\ast)$ goes beyond that by claiming that if a statement is consistent, then it is already true.
Here, the statement in question needs to come from a restricted class of statements, and "consistent" needs to mean "consistent in a strong sense."
It turns out that $(\ast)$ is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory.
This is joint work with D. Asperó building upon earlier work of R. Jensen and (ultimately) Keisler's "consistency properties."
Sept. 29, 2020
Monroe Eskew :
11 a.m. in Zoom
Abstract
The motivating question for this work is: Can we have both a saturated ideal and the tree property on $\aleph_2$? Towards the negative direction, we show that for a regular cardinal $\kappa$, if $2^{<\kappa}\leq\kappa^+$ and there is a weakly presaturated ideal on $\kappa^+$ concentrating on cofinality $\kappa$, then $\square^*_\kappa$ holds. This proves a conjecture of Foreman about the approachability ideal on $\aleph_2$ under the assumption that the continuum is at most $\aleph_2$. A surprising corollary is that if there is a weakly presaturated ideal $J$ on $\aleph_2$ such that $P(\aleph_2)/J$ is a proper forcing, then CH holds. This is joint work with Sean Cox.
Oct. 13, 2020
Matteo Viale :
2 p.m. in Zoom
Abstract
We show that (assuming large cardinals) set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic definable concepts of second and third order arithmetic, and appealing to the model-theoretic notions of model completeness and model companionship.
Specifically we develop a general framework linking generic absoluteness results to model companionship and show that (with the required care in details) a $\Pi_2$-property formalized in an appropriate language for second or third order number theory is forcible from some $T$ extending ZFC + large cardinals if and only if it is consistent with the universal fragment of $T$ if and only if it is realized in the model companion of $T$.
Part (but not all) of our results are conditional to the proof of Schindler and Asperò that Woodin’s axiom $(\ast)$ can be forced by a stationary set preserving forcing.
Oct. 26, 2020
Jenna Zomback :
4 p.m. in Zoom
Abstract
A pointwise ergodic theorem for the action of a transformation $T$ on a probability space equates the global property of ergodicity of the transformation to its pointwise combinatorics. Our main result is a backward (in the direction of $T^{-1}$) ergodic theorem for countable-to-one probability measure preserving (pmp) transformations $T$. We discuss various examples of such transformations, including the shift map on Markov chains, which yields a new (forward) pointwise ergodic theorem for pmp actions of finitely generated countable groups, as well as one for the (non-pmp) actions of free groups on their boundary. This is joint work with Anush Tserunyan.
Nov. 3, 2020
James Freitag :
3 p.m. in Zoom
Abstract
We will give an overview of the construction of binding groups in model theory, also discussing their history and applications. Following this, we will talk about a new invariant in geometric stability theory with very concrete applications. It turns out the key to understanding this invariant has to do with understanding just how transitive a binding group can be. We will demonstrate the ideas in several settings (finite Morley rank, o-minimality, algebraic groups, differential algebraic equations) and leave some time for discussion speculating about other possible settings in which the notions might be useful.
Nov. 10, 2020
Vincenzo Mantova :
2 p.m. in Zoom
Abstract
Surreal numbers are a common generalisation of real and ordinal numbers, where one replaces Dedekind completeness of the reals with saturation, and the well-order of ordinals with a well-founded binary tree-like partial order called "simplicity relation". Using simplicity, one can define canonical, "simplest" operations of sum and product that generalise (Hessenberg) sum and product of real and ordinals, as well as simplest restricted analytic functions, global exponentiation, and H-field derivations with real kernel. I will present compact definitions for most of those based on simplicity, bypassing the traditional recursive definitions of Conway.
While the above functions turn out to be nonstandard models of the corresponding ones on the reals (and on LE-series for derivation with exp), the simplest *integer* part, the ring of the so-called omnific integers, presents other challenges. The (positive part of the) structure is a model of open induction but is otherwise far from being a nonstandard model of the integers. Conway conjectured that omnific integers satisfy at least a suitable weakening of unique factorisation. This is still unsolved: I will discuss the few known results, many deriving from work of Berarducci, and the latest joint work with L'Innocente, that can be obtained via certain valuations.
Nov. 24, 2020
Rachel Skipper :
2 p.m. in Zoom
Abstract
The space of subgroups of a group has a natural Polish topology and understanding this space can help to understand the group.
In this talk, we will consider the Cantor-Bendixson derivative and rank for the space of subgroups of the Grigorchuk group, using it to stratify the subgroups of this group.
Jan. 19, 2021
Filippo Calderoni :
4 p.m. in Zoom
Abstract
In this informal talk we will discuss an anticlassification result for Archimedean ordered groups of finite rank recently obtained in joint work with Marker, Motto Ros, and Shani. We will explain how this is connected to some interesting open questions about the theory of countable Borel equivalence relations.
Feb. 2, 2021
Sean Cox :
4 p.m. in Zoom
Abstract
The class of projective modules is central to classical
homological algebra. Relative homological algebra attempts to use
some class $\mathcal{G}$ and "do" homological algebra, but with
$\mathcal{G}$ playing the same role that the class of projectives
played in the original setting. However, an essential requirement for
this to work is that $\mathcal{G}$ be a "precovering" class (also
called a "right-approximating" class) of modules. There has been
considerable work in the last 20 years on the question of whether the
class of "Gorenstein Projective" modules is always a precovering class
(over every ring). While the question remains open, it is now known
that the answer is affirmative if there are enough large cardinals in
the universe. This was first proved by Saroch, and then
(independently) by me, using entirely different methods. I will
discuss some of the key ideas of my proof, especially the use of
(set-theoretic) "elementary submodel" arguments and Stationary Logic.
Feb. 16, 2021
Dino Rossegger :
4 p.m. in Zoom
Abstract
We present new results on the complexity of the classification problem of countable structures and their computational complexity. We show that the elementary bi-embeddability relation on the class of graphs is analytic complete under Borel reducibility by giving a reduction from the bi-embeddability relation on graphs. We then compare the degree spectra with respect to these equivalence relations. The degree spectrum of a countable structure with respect to an equivalence relation $E$ is the set of Turing degrees of structures $E$ equivalent to it. We show that the degree spectra of structures with respect to bi-embeddability and elementary bi-embeddability are related: Every bi-embeddability spectrum of a graph is the set of jumps of Turing degrees in the elementary bi-embeddability spectrum of a graph.
March 2, 2021
Katrin Tent :
11 a.m. in Zoom
Abstract
In joint work with Segal we use the fact that for Chevalley groups $G(R)$ of rank at least $2$ over a ring $R$ the root subgroups are (nearly always) the double centralizer of a corresponding root element to show under mild restrictions on the ring $R$ that $R$ and $G(R)$ are bi-interpretable. (This holds in particular for any field $k$.) For such groups it then follows that the group $G(R)$ is finitely axiomatizable in the appropriate class of groups provided $R$ is finitely axiomatizable in the corresponding class of rings.
March 9, 2021
Natasha Dobrinen :
4 p.m. in the internet
Abstract
Analogues of the infinite Ramsey theorem to infinite structures have been studied since the 1930’s, when Sierpinski gave a coloring of pairs of rationals into two colors such that, in any subset of the rationals forming a dense linear order, both colors persist. Such a coloring is called “unavoidable” since both colors persist in any infinite substructure isomorphic to the original (in this case the rationals as a linear order). In the 1970’s Galvin showed that two is the optimum number for pairs of rationals, while Erdos, Hajnal and Posa extended Sierpinski’s result to colorings of edges in the Rado graph. These results instigated a steady stream of results for the next several decades, a pinnacle of which was the work of Laflamme, Sauer, and Vuksanovic finding the exact number of colors for unavoidable colorings of finite graphs inside the Rado graph, as well as other Fraisse structures with finitely many binary relations, including the generic tournament. This exact number is called the “big Ramsey degree”, a term coined by Kechris, Pestov, and Todorcevic.
In this talk, we will provide a brief overview of the area of big Ramsey degrees on Fraisse limits. Then we will present recent joint work with Rebecca Coulson and Rehana Patel characterizing the big Ramsey degrees for some seemingly disparate Fraisse classes. We formulate an amalgamation property, which we call the Substructure Free Amalgamation Property, and show that every Fraisse relational class with finitely many relations satisfying SFAP has big Ramsey degrees which are characterized in a manner as simply as those of the Rado graph. A more general property for disjoint amalgamation classes, which we call SDAP^+, also ensures the same simple characterization of big Ramsey degrees. One of the novelties of our approach is that we build trees of quantifier-free 1-types with special nodes coding the vertices in a given enumerated Fraisse limit. Then we use the method of forcing to do an unbounded search for a finite object, which produces in ZFC the exact big Ramsey degrees for these structure. SDAP^+ holds for unrestricted relational structures, relational structures with forbidden 3-irreducible substructures, and others, producing new lines of results while recovering in a streamlined manner several previous results, including those of Laflamme, Sauer, and Vuksanovic.
If you are interested in attending the seminar and joining the mailing list, please write jfreitag@uic.edu an email.
Gianluca Basso :
11 a.m. in Zoom
Abstract
The goal of projective Fraïssé theory is to approximate compact metrizable spaces via classes of finite structures and glean topological or dynamical properties of a space by relating them to combinatorial features of the associated class of structures. We will discuss general results, using the framework of compact metrizable structures, as well as applications to the study a class of one-dimensional compact metrizable spaces, that of smooth fences, and to a particular smooth fence with remarkable properties, which we call the Fraïssé fence.
March 16, 2021
Alexi Block Gorman :
4 p.m. in Zoom
Abstract
Büchi automata are the natural extension of finite automata, also called finite-state machines, to a "machine" that accepts infinite-length inputs. We say a subset X of the reals is r-regular if there is a Büchi automaton that accepts (one of) the base-r expansions of every element in X, and rejects the base-r expansion of each element in its complement. We can analogously define r-regular subsets of higher arities of the reals, and these sets often exhibit fractal-like behavior--e.g., the Cantor set is 3-regular. There are several known--and remarkable--connections in logic to Büchi automata, including the fact that the expansion of the real additive group by every r-regular subset of [0,1] for some fixed positive integer r interprets the monadic second-order theory of the natural numbers with successor. In this talk, I will focus on some of the geometric behavior of closed r-regular set in terms of fractal dimensions, and discuss how closed r-regular sets with and without integer Hausdorff dimension form a dichotomy in terms of first order definability in expansions of the real additive group by a predicate for a specific r-regular set.
March 30, 2021
Sittinon Jirattikansakul :
4 p.m. in Zoom
Abstract
Prikry-type forcings have been developed to tackle problems on singular cardinals. A specific example is the Singular Cardinal Hypothesis (SCH): the size of a powerset of a singular cardinal is the smallest value consistent with ZFC. In particular, if $\theta$ is a singular strong limit cardinal, then $2^\theta=\theta^+$. In 2019, Gitik developed a forcing that violates the SCH, given that the target cardinals were singular in a ground model. In this talk, we will discuss the Gitik's forcing with interleave collapses, to violate the SCH at the small alephs.
April 6, 2021
Raphael Carroy :
11 a.m. in Zoom
Abstract
What is a finite basis result for a quasi-order? A quasi-order is a transitive and reflexive relation on a set (or a class). Given a quasi-order $\leq_Q$ on a set $Q$ and a subset $A$ of $Q$, a basis for $A$ is a subset $B$ of $A$ such that for all $a \in A$ there exists $b \in B$ so that $b \leq_Q a$. The quasi-order $\leq_Q$ has a symmetrization: $p \equiv_Q q$ if and only if $p \leq_Q q$ and $q \leq_Q p$, which is an equivalence relation. We say that the basis $B$ is finite if its quotient by $\equiv_Q$ is finite.
We consider the existence of a morphism between two structures in a given class as a quasi-order on the class of structures. I will talk about some finite basis results on classes of graphs and classes of functions for various notions of morphisms, and the interplay between them.
April 13, 2021
Andrew Marks :
4 p.m. in Zoom
Abstract
We characterize which Borel functions are decomposable into
a countable union of functions which are piecewise continuous on
$\Pi^0_n$ domains, assuming projective determinacy. One ingredient of
our proof is a new characterization of what Borel sets are $\Sigma^0_n$
complete. Another important ingredient is a theorem of Harrington that
there is no projective sequence of length $\omega_1$ of distinct Borel
sets of bounded rank, assuming projective determinacy. This is joint
work with Adam Day.
April 20, 2021
Hang Lu Su :
11 a.m. in Zoom
Abstract
Left-orderable groups are groups which admit a strict total order $\prec$ which is invariant under left-multiplication, called left-order. I will explain our approach to the problem of algorithmically determining whether $g \prec h$ using the framework of formal languages, and attempt to give an intuitive justification to the arbitrary choices made in formalising our study of this problem. Finally, I will give an overview of our results concerning left-orders of low computational complexity. Some of these results are joint work with Yago Antolín and Cristóbal Rivas.
April 27, 2021
Gabriel Goldberg :
4 p.m. in Zoom
Abstract
I'll discuss some ideas relating strong compactness, club filters, and inner models, with two applications: first, a generalization of Woodin's HOD dichotomy that applies to any inner model with access to the \(\omega\)-club filter on each ordinal of uncountable cofinality, and second, an analysis of strong compactness in the HODs of models satisfying either the Axiom of Determinacy or choiceless large cardinal assumptions.
Aug. 31, 2021
Matt Devilbiss :
4 p.m. in 636 SEO
Abstract
In this talk, I will outline a new technique for showing that nonlinear algebraic differential equations are strongly minimal. This is used to prove the strong minimality of generic differential equations with sufficiently large degree, answering a question of Poizat (1980). Time permitting, I will also discuss ongoing work in applying this method to differential equations of interest whose coefficients are not generic. This is joint work with James Freitag.
Sept. 7, 2021
Dima Sinapova :
4 p.m. in 636 SEO
Abstract
Two classical results of Magidor are:
(1) from large cardinals it is consistent to have reflection at $\aleph_{\omega+1}$, and
(2) from large cardinals it is consistent to have the failure of the singular cardinal hypothesis (SCH) at $\aleph_\omega$.
These principles are at odds with each other. The former is a compactness type principle. (Compactness is the phenomenon where if a certain property holds for every smaller substructure
of an object, then it holds for the entire object.) In contrast, failure of SCH is an instance of incompactness. The natural question is whether we can have both of these simultaneously.
We show the answer is yes
This is joint work with Alejandro Poveda and Assaf Rinot.
Sept. 14, 2021
James Freitag :
4 p.m. in 636 SEO
Abstract
In recent years, definable sets in o-minimal geometry have seen widespread application in number theory around diophantine geometry and transcendence. The diophantine applications of o-minimality center around various versions of results for counting points of bounded height originating from the work of Pila and Wilkie. In the full generality of o-minimal geometry, the results of Pila and Wilkie can not be improved (it is not even clear what this would mean), but various improvements which are vital in applications have been obtained recently.
One of the chief settings in which there is an improved counting theorem is the setting in which the functions are assumed to be Pfaffian, a setting introduced by Khovanskii.
A natural open problem is wether every differential algebraic function interpretable in an o-minimal structure is Pfaffian? We will answer this open question. Our analysis answers a number of recent open questions of Binyamini and Novikov, Bianconi, Armitage.
Sept. 21, 2021
No seminar :
4 p.m. in 636 SEO
Sept. 28, 2021
Minh Tran :
4 p.m. in 636 SEO
Abstract
A recurring theme in model theory is that under suitable assumptions (stabilty, NIP, small expansions, etc), definable groups resemble geometric groups (algebraic groups, Lie groups, etc) at different levels. I will explain how this comes into play in the recent solution of the Inverse Kemperman problem by Jinpeng An, Yifan Jing, Ruixiang Zhang and I. Our proof also uses some descriptive set theory, in particular, an automatic continuity result.
Oct. 12, 2021
Clinton Conley :
4 p.m. in 636 SEO
Abstract
We say that a subset A of the sphere r-divides it if r-many rotations
of A perfectly tile the sphere's surface. Such divisions were first
exhibited by Robinson ('47) and developed by Mycielski ('55). We
discuss a colorful approach to finding these divisions which are
Lebesgue measurable or possess the property of Baire. This includes
joint work with J. Grebik, A. Marks, O. Pikhurko, and S. Unger.
Oct. 19, 2021
Leo Jimenez :
3 p.m. in 636 SEO
Abstract
In geometric stability theory, the duality between locally modular and not plays a central structural role. For example, in many important cases, non-local modularity implies the interpretability of a field, a phenomenon known as Zilber's dichotomy. Hrushovski's classical counterexample to this behavior, while not locally modular, still has a relatively rudimentary forking geometry: it is CM-trivial, which prevents the interpretability of a field.
More recently, a relative generalization of local modularity, inspired by the behavior of compact complex spaces, was defined: the Canonical Base Property (CBP). It was shown to hold in many key structures, for example DCF0, where it was used by Pillay and Ziegler to show Zilber's dichotomy. It was first conjectured that the CBP held for all superstable finite rank structures, until Hrushovski, Palacín and Pillay produced the first counterexample, as a reduct of an algebraically closed field of characteristic zero. Since then, all counterexamples produced involved a field, and it is natural to ask if this is necessary.
In this talk, I will answer this question negatively by presenting a CM-trivial structure without the CBP. Joint with Thomas Blossier.
Neer Bhardwaj :
4 p.m. in 636 SEO
Abstract
I’ll give an account of the Pila-Wilkie counting theorem and some of its extensions and generalizations. We exploit semialgebraic cell decomposition more thoroughly to simplify the deduction from the main ingredients of the original proof. Only very basic knowledge of o-minimality will be assumed; this is joint work with Prof. Lou van den Dries.
Oct. 26, 2021
Thomas Gilton :
4 p.m. in 636 SEO
Abstract
A broad question guiding much contemporary research in set theory is the extent to which the set-theoretic universe resembles certain "inner models" such as Gödel's constructible universe L. One way of cashing out this resemblance is the extent to which certain Rigidity principles, such as Jensen's square principles, hold in the model of interest. Many of these rigidity principles imply the failure of other combinatorial principles of interest, ones which exhibit a sizable amount of "reflection" or "compactness," and of which the Tree Property and Stationary Reflection are examples. We thus have two classes of interesting combinatorial principles (rigidity on the one hand, and compactness/reflection on the other), instances of which are often inconsistent.
Given this tension, a fruitful line of contemporary research investigates when instances from these classes are jointly consistent. In this talk, we will discuss a joint result of the speaker with Omer Ben-Neria which contributes to the study of this tension; our result is that Club Stationary Reflection is consistent with the Special Aronszajn Tree Property on \omega_2. After briefly surveying "rigidity and compactness", we will discuss the main obstacles to obtaining our result, focusing on how the above-mentioned tension arises. Then we outline the main tools for overcoming these problems, namely, our notions of posets which are Strongly Proper or Completely Proper with respect to the weakly compact filter. As time permits, we will also discuss our new preservation theorems for stationary sets and Aronszjan trees.
Nov. 2, 2021
Dana Bartošová :
4 p.m. in 636 SEO
Abstract
In the past two decades there has been much investigation into structural Ramsey theory, that is, natural generalizations of the finite, and more recently infinite, Ramsey's theorem to finite/countable first order structures, such as graphs, hypergraphs, Boolean algebras, or vector spaces over finite fields. We investigate how the Ramsey phenomena transport from a countable class of finite structures to its ultraproducts. This is a joint work in progress with Mirna Džamonja, Rehana Patel, and Lynn Scow.
Nov. 9, 2021
Sebastian Eterovic :
4 p.m. in 636 SEO
Abstract
The Existential Closedness (EC) problems are natural questions about the algebraic properties of important transcendental functions in arithmetic geometry. These problems have their origins in model theory and can be seen as analogues of Hilbert's Nullstellensatz for certain systems of analytic equations, and also as counterparts of Schanuel-type conjectures. In the talk I will introduce the EC problems, I will explain the partial results that have been proven so far, and also how they fit in a bigger picture with other important theorems and conjectures in arithmetic geometry.
Nov. 23, 2021
Iian Smythe :
4:15 p.m. in 636 SEO
Abstract
In the late 90's, Gowers proved a Ramsey-theoretic dichotomy for subspaces of infinite-dimensional Banach spaces. The combinatorial essence of this result was later extracted by Rosendal in the setting of discrete vector spaces. Both dichotomies say, roughly, that given an analytic partition of the set of infinite block sequences of vectors, there is an infinite-dimensional subspace with a wealth of block sequences entirely contained in, or disjoint from, one piece of the partition. We will describe a new "parametrized" form of Rosendal's dichotomy: Given an analytic family of partitions indexed by the reals, there is a single subspace which witnesses Rosendal's dichotomy for uncountably many of the partitions, simultaneously. An integral part of our proof is the preservation of certain analogues of selective ultrafilters, by Sacks forcing. We will also discuss applications to families of linear transformations.
Nov. 30, 2021
Denis Osin :
4:15 p.m. in 636 SEO
Abstract
The space of finitely generated marked groups, denoted by $\mathcal G$, is a locally compact Polish space whose elements are groups with fixed finite generating sets; the topology on $\mathcal G$ is induced by local convergence of the corresponding Caley graphs. I will describe a necessary and sufficient condition for a closed subspace $\mathcal S\subseteq \mathcal G$ to satisfy the following zero-one law: for any sentence $\sigma$ in the infinitary logic $\mathcal L_{\omega_1, \omega}$, the set of all models of $\sigma$ in $\mathcal S$ is either meager or comeager. In particular, the zero-one law holds for certain subspaces associated to hyperbolic groups. This leads to the following (somewhat unexpected) corollary: generic limits of non-cyclic, torsion-free, hyperbolic groups are elementarily equivalent. We will discuss other applications and open problems.
Feb. 1, 2022
James Freitag :
4 p.m. in 636 SEO
Abstract
We give a surprising theorem about functional transcendence and solutions of differential equations.
Feb. 8, 2022
Dima Sinapova :
4 p.m. in 636 SEO
Feb. 22, 2022
Filippo Calderoni :
4 p.m. in 636 SEO
Abstract
The theory of countable Borel equivalence relations (CBERs) analyzes the actions of countable groups. The main question is how much information is encoded by the orbit space. The more information is encoded the more rigid the action is.
We prove rigidity results for the action of the group of rational rotations on spheres in higher dimension. This connects to superrigidity in work by Margulis and to Zimmer's program about the actions of discrete subgroups of Lie groups on manifolds. Moreover, our methods provide new examples of CBERs, and a new proof of a fundamental theorem of Adams and Kechris about Borel complexity.
March 8, 2022
David Marker :
4 p.m. in 636 SEO
Abstract
Using ideas from geometric stability theory we construct differentially closed fields with no nontrivial automorphims.
March 15, 2022
Matt Foreman :
4 p.m. in 636 SEO
Abstract
In 1967 Smale proposed classifying the "qualitative behavior” of diffeomorphisms of compact smooth manifolds. Despite very significant activity, the general problem remained open. However joint work with Gorodetski shows that in dimensions 2 and above, there are no complete invariants and in dimension 5 and above the equivalence relation is not even Borel. Hence there can be no classification using inherently countable information.
March 29, 2022
Artem Chernikov :
4 p.m. in 636 SEO
Abstract
A randomization of a first-order structure M, introduced by Keisler, is a structure M^R in continuous logic whose elements are the "random" elements of M. One can think of it as a continuous structure whose types correspond to probability measures on the space of types of the original structure. Randomization preserves certain model-theoretic tameness properties, e.g. stability and NIP. The latter was demonstrated by Ben Yaacov via developing aspects of the VC-theory (Vapnik-Chervonenkis) in the continuous setting, connected to earlier work of Talagrand and others. A more general hierarchy of n-dependent theories was introduced by Shelah, with the case n=1 corresponding to NIP: a theory is n-dependent if the edge relation of an infinite generic (n+1)-hypergraph is not definable. We will discuss n-dependence in continuous logic and demonstrate that n-dependence is also preserved by Keisler randomization: the main point is that the average of a family of uniformly n-dependent functions is n-dependent. Our proof relies on structural Ramsey theory and multidimensional de Finetti-type results (and provides in particular a new proof in the NIP case). Joint work with Henry Towsner.
April 5, 2022
David Meretzky :
4 p.m. in 427 SEO
Abstract
Picard-Vessiot theory is well behaved over differential fields with algebraically closed fields of constants. Under this assumption PV extensions for an ordinary linear homogeneous differential equation exist and are unique inside of a fixed differential closure. For general fields of constants, one no longer has uniqueness and there is an ``indirect" Galois correspondence. We describe the more limited Galois correspondence for differential extensions which are generated by a fundamental system and possibly allow some new constants. We discuss the general model theoretic context.
April 12, 2022
John Baldwin :
4 p.m. in 636 SEO
Abstract
A scaffold for mathematics includes both local foundations for various areas of
mathematics and productive guidance in how to unify them. In a scaffold the unification does not take place by a common axiomatic basis but consists of a systematic
ways of connecting results and proofs in various areas of mathematics. Two scaffolds, model theory and category theory, provide local foundations for many areas
of mathematic including two flavors (material and structural) of set theory and
different approaches to unification. We will discuss salient features of the two scaffolds including their contrasting but bi-interpretable set theories. We focus on the
contrasting treatments of ‘size’ in each scaffold and the advantages/disadvantages
of each for different problems.
April 26, 2022
John Baldwin :
4 p.m. in 636 SEO
Abstract
Combinatorics:
We introduce a uniform method
of proof for the following results. For each of the following
conditions, there are $2^{\aleph_0}$ families of Steiner systems, satisfying
that condition: i) Theorem 1: (extending Chicot et al) each Steiner triple
system is $\infty$-sparse and has a uniform but not perfect path graph; ii)
(Theorem 2: (extending Cameron-Webb) each Steiner $k$-system (for $k=p^n$) is
$2$-transitive and has a uniform path graph (infinite cycles only); iii)
Theorem 3: (extending Fujiwara), each is anti-Pasch (anti-mitre); iv) Theorem
4 Steiner $k$-system has an explicit quasi-group structure. In each case all
members of the family satisfy the same complete strongly minimal theory and
it has $\aleph_0$ countable models and one model of each uncountable
cardinal. Item iii) is particularly interesting since it fails completely
for the most obvious construction of such Steiner systems, which fall in 1)
of the classification below.
Classification: We will briefly discuss the lengthy proof (with Verbovskiy) that the strongly minimal Steiner systems and Hrushovski's
original example a) do not admit a) elimination of imaginaries or (more
strongly) b) an $\emptyset$-definable binary function. This result justifies
the observation that changing the $\mu$-function or adding axioms like linear
space yields profoundly different strongly minimal sets. The ab initio
Hrushovski construction yields
non-trivial flat classes which split into those
i) with
no definable binary function, ii)
definable binary functions exist. These can be further subdivided as: a) no
commutative binary function (elimination of imaginaries fails); b) strongly
minimal quasigroups discussed in the first part of the
talk;
c) non-Desarguesian projective planes coordinatized by ternary fields.
Aug. 30, 2022
Tom Benhamou :
4 p.m. in 636 SEO
Abstract
We present a property of filters discovered by F. Galvin which he proved to hold for normal filters over strongly regular cardinals, and gained renewed in-
terest due to recent developments in set theory. In the first part of the talk, we will provide applications of this property to infinite combintorics and to Prikry type
forcing. The second goal will be to present some strengthening of Galvin’s theorem, and prove that in some canonical inner models, every $\kappa$-complete ultrafilter over $\kappa$ has the Galvin property. We will also present constructions of filters and ultra-filters without the Galvin property and prove a new result answering a question of
Garti, Shelah and B. about the existence of such ultrafilters on very large cardinals.
In the third part of the talk, we continue the work of U.Abraham and S.Shelah
who produced a model where the club filter fails to satisfy the Galvin property in a
strong sense at $\kappa ^+$, where $\kappa$ is a regular cardinal and $2^ \kappa > \kappa ^+$.
We will produce
a model where the club filter fails to satisfy the Galvin property at $\kappa ^+$, where $\kappa$ is singular and $2^\kappa > \kappa ^+$. We will obtain this model from the optimal large cardinal
assumptions and explore the possibility of obtaining the stronger form of failure as
in the Abraham and Shelah model. This is a joint work with M.Gitik, S.Garti and
A.Poveda.
Sept. 6, 2022
Thomas Kucera :
4 p.m. in 636 SEO
Abstract
Joint work with Philipp Rothmaler, CUNY.
Elementary duality of positive primitive formulas over left and right modules is a model-
theoretic tool (introduced by Prest [1988] and developed extensively by I. Herzog [1993]) in the
context of the finitary first order model theory of modules that relates the category of left R-
modules to the category of right R-modules, and much more. Prest, Rothmaler, and Ziegler
[1994] extended the basic ideas to certain infinitary analogues of ppfs, (with finitely many free
variables).
Rothmaler and I extend these results further, to positive primitive properties of infinite sequences
of elements in a module. We identify the syntactic forms of the dualizable properties, how to
compute the duals, and the two kinds of infinitary existential quantifier that arise. We show that a
certain class of modules, the locally projective modules of Zimmermann-Huisgen [1976], has a
straightforward axiomatization by implications of dualizable formulas of our kind (but not likely
by the formulas of [PRZ]). The elementary dual theory is easy to describe formally; but the
algebraic content of it remains stubbornly obscure.
I will give a general overview of this work, while avoiding most of the very technical machinery
underlying it.
Sept. 27, 2022
Nigel Pynn-Coates :
4 p.m. in 636 SEO
Abstract
Let T be a complete, model complete, power bounded o-minimal theory extending the theory of real closed fields. A T-convex T-differential field is an expansion of a model of T by a valuation and a derivation, each of which is compatible with the o-minimal structure, the former in the T-convex sense of van den Dries--Lewenberg and the latter in the sense of Fornasiero--Kaplan. When T is the theory of the real field with restricted analytic functions, we can expand an ordered differential Hahn field to a T-convex T-differential field, in which case the derivation is monotone, i.e., weakly contractive with respect to the valuation (monotone differential Hahn fields were studied earlier by Scanlon and Hakobyan). I will describe joint ongoing work with Kaplan on monotone T-convex T-differential fields, achieving among other results an Ax--Kochen/Ershov type theorem for such structures. A key step is isolating an appropriate analogue of henselianity in this setting.
Oct. 11, 2022
Will Adkisson :
4 p.m. in 636 SEO
Abstract
The strong tree property and ITP (also called the super tree property) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility in much the same way that the tree property characterizes weak compactness. That is, an inaccessible cardinal $\kappa$ is strongly compact if and only if the strong tree property holds at $\kappa$, and supercompact if and only if ITP holds at $\kappa$.
Generalizing a result of Neeman about the tree property, we show that it is consistent for ITP to hold at $\aleph_n$ for all $1< n < \omega$ simultaneously with the strong tree property at $\aleph_{\omega+1}$. We also show that it is consistent for ITP to hold at $\aleph_n$ for all $3 < n < \omega$ and at $\aleph_{\omega+1}$ simultaneously. Finally, turning our attention to singular cardinals of uncountable cofinality, we show that it is consistent for the strong and super tree properties to hold at successors of singulars of multiple cofinalities simultaneously.
Oct. 18, 2022
Scott Mutchnik :
4 p.m. in 636 SEO
Abstract
Model theory has been described as a "geography of tame mathematics," creating a map of the universe of first-order theories according to various dividing lines, such as tree properties or order properties. While some regions of this map, such as the stable theories or simple theories, are well-understood to varying degrees, as we progress outward it even becomes open whether some regions are empty or not. Extending the NSOP_n hierarchy of Shelah [1995] defining an ascending chain of strong order properties for n > 2, Džamonja and Shelah [2004] introduce two further tree properties, NSOP_1 and NSOP_2, and ask whether the implications between NSOP_1 and NSOP_2 and between NSOP_2 and NSOP_3 are strict. We have answered the first of these questions, showing that the class NSOP_1 coincides with NSOP_2. We discuss this result and some aspects of its proof, which incorporates ideas from various other regions of the model-theoretic map such as the NSOP_1, NSOP_3 and NTP_2 theories.
Oct. 25, 2022
Kevin H Zhou :
4 p.m. in 636 SEO
Abstract
Learning automata by queries is a long-studied area of problems with many applications in AI, automatic verification, model checking, and more. In this setting, a learner plays a game with an oracle, where the goal is to identify some unknown target automaton by interactively submitting queries to the oracle. The study of query learning of automata was initiated by Angluin in 1987 with the introduction of the L* algorithm that learns deterministic finite automata (DFAs) with a polynomial number of queries, and most subsequent work has focused on adapting the L* algorithm to different settings. More recently, Chase and Freitag used ideas from model theory to develop query learning bounds in terms of the Littlestone and consistency dimensions, and apply this to the setting of regular languages to obtain qualitatively different results compared to Angluin. We extend this work, applying the method to two generalizations of DFAs: 1) advice DFAs, where the automaton has access to a fixed advice string, and 2) nominal DFAs, a generalization of DFAs to infinite alphabets and state sets. In the setting of advice DFAs, we obtain the first known query learning results, while in the setting of nominal DFAs, we obtain significant improvements over prior work.
Nov. 1, 2022
Nick Ramsey :
4 p.m. in 636 SEO
Abstract
A theory T is called binary if any two tuples have the same type if and only if all corresponding subtuples of length 2 have the same type. With this strong restriction on theories, it turns out that certain classification-theoretic dividing lines collapse: for example, we show that a binary NSOP_1 theory is simple, and a binary NSOP_3 theory is NTP_1. Motivated by these results, we develop the basics of neostability theory for the broader category of treeless theories. We show such theories come equipped with a natural notion of independence, defined in terms of generically stable partial types, which is meaningful in both simple and NIP theories. This is joint work with Itay Kaplan and Pierre Simon.
Nov. 8, 2022
Isaac Goldbring :
4 p.m. in Zoom
Abstract
In this talk, we discuss existentially closed measure preserving actions of countable groups. A classical result of Berenstein and Henson shows that the model companion for this class exists for the group of integers and their analysis readily extends to cover all amenable groups. Outside of the class of amenable groups, relatively little was known until recently, when Berenstein, Henson, and Ibarlucía proved the existence of the model companion for the case of finitely generated free groups. Their proof relies on techniques from stability theory and is particular to the case of free groups. In this talk, we will discuss the existence of model companions for measure preserving actions for the much larger class of universally free groups (also known as fully residually free groups), that is, groups which model the universal theory of the free group. We also give concrete axioms for the subclass of elementarily free groups, that is, those groups with the same first-order theory as the free group. Our techniques are ergodic-theoretic and rely on the notion of a definable cocycle. This talk represents ongoing work with Brandon Seward and Robin Tucker-Drob.
Nov. 15, 2022
Alejandro Poveda :
4 p.m. in 636 SEO
Abstract
In this talk I will report on the $\Sigma$-Prikry framework developed with Rinot and Sinapova. The main goal will be to describe an application of this framework to the study of stationary reflection at the level of singular cardinals. More precisely, I will describe how to get a model where GCH holds up to $\aleph_\omega$, SCH fails at $\aleph_{\omega+1}$ and stationary reflection holds at $\aleph_{\omega+1}$. This answers a question by Magidor from 1982. If time permits I will comment on some current work in progress.
Nov. 29, 2022
Spencer Unger :
4 p.m. in Zoom
Abstract
Combining elements from a long line of research on the tree property, we aim to prove that it is consistent that every regular cardinal between aleph_2 and aleph_{omega^2+3} has the tree property while aleph_{omega^2} is strong limit. In this talk, I'll give some background and give a sampling of some of the many ideas that go into the proof along with their connections to other research. This is joint work in progress with James Cummings, Yair Hayut, Menachem Magidor, Itay Neeman and Dima Sinapova.
Jan. 24, 2023
John T. Baldwin :
4 p.m. in 636 SEO
Abstract
Sacks’ student Lenore Blum provided the first intuitive axiomatization of the
theory of differentially closed fields. Moreover, she established the $\omega$-stability of the
theory and thus the uniqueness of the prime model. Sacks book, Saturated Model
Theory, was the first book expounding stability theory and presented the notion of a
differentially closed field to a general logical audience. Zilber initiated the program
of proving $L_{\omega_1,\omega}$-categoricity in power for ‘canonical mathematical structures’. In
this talk we will describe recent advances unifying these two programs.
Jan. 31, 2023
James Freitag :
4 p.m. in 636 SEO
Abstract
We'll talk about some problems from Hrushovski's paper Almost Orthogonal Regular Types in the context of differential fields and omega stable theories.
Feb. 7, 2023
Tom Benhamou :
4 p.m. in 636 SEO
Abstract
In this talk, we will focus on certain saturation properties of filters and ultrafilters which generalizes the so-called Galvin property. In the first part of the talk, we will present a connection between such ultrafilters and the existence of Slim-Kurepa trees. We will then present several results regarding the existence of non-Galvin ultrafilters under several large cardinal assumptions. Finally, if time permits, we will present a recent application to canonical inner models and some open related questions.
Feb. 14, 2023
Christian Schulz :
4 p.m. in 636 SEO
Abstract
Let k, l ≥ 2 be two multiplicatively independent integers. It is a well-known implication of Büchi that the expansion of (N, +) by any k-automatic relation has a decidable theory. But as shown by Bès in 1997, there exist k-automatic sets S_k such that for any l-automatic set S_l, the structure (N, +, S_k, S_l) defines multiplication. Here we show that this dichotomy does not extend to all expansions of (N, +) by k-automatic and l-automatic sets: the structure (N, +, k^N, l^N) does not have a decidable theory, nor does it define multiplication.
Feb. 21, 2023
Gabriel Goldberg :
4 p.m. in 636 SEO
Abstract
The ultrapower axiom (UA) is a set theoretic principle abstracted from inner model theory. By design, UA is true in all known canonical inner models of set theory, and it is likely to be true in any model constructed by anything like the current methodology of inner model theory. The consistency of UA with large cardinal axioms at the level of a supercompact therefore provides an important test question and guide in the attempt to generalize inner model theory to this level. In this talk I'll outline the ramifications of UA in large cardinal theory, which turn out to be plentiful, and explain some indirect ways that this theory has helped to answer some questions in set theory unconditionally.
March 7, 2023
Jing Zhang :
4 p.m. in 636 SEO
Abstract
We will discuss and address some problems concerning a family of weakenings of the usual Ramsey theorem. Instead of asking for a monochromatic complete graph, very roughly speaking, these versions ask for variations of a monochromatic topological copy of the complete graph, i.e. each edge is replaced by a path. This allows the possibility of consistency at small uncountable cardinals or the continuum where the usual Ramsey theorem is inconsistent. We will also say something about motivation and some recent applications. Joint work with Hrusak and Shelah.
March 14, 2023
Marcos Mazari Armida :
4 p.m. in 636 SEO
Abstract
The classical model-theoretic notion of superstability was introduced by Shelah in the late sixties for first-order theories and in the late nineties for abstract elementary classes. In this talk we will show that superstability is a natural algebraic property by characterizing some classical classes of rings, such as noetherian rings and perfect rings, via superstability of certain classes of modules.
March 28, 2023
Joel D. Hamkins :
4 p.m. in 636 SEO
April 4, 2023
Dima Sinapova :
4 p.m. in 636 SEO
Abstract
The tree property is an uncountable analogue of Konig's infinity lemma. At a cardinal $\kappa$, it states that every tree of height $\kappa$ and levels of size less than $\kappa$ has a cofinal branch. At $\aleph_1$ the tree property fails, and at $\aleph_2$ the tree property is equiconsistent with a weakly compact cardinal (Mitchell; Silver, 1972). Going further, in 1983 Abraham showed the tree property can hold simultaneously at $\aleph_2$ and $\aleph_3$, using a much stronger large cardinal hypothesis -- a supercompact cardinal.
Since then, it has been a long standing project in set theory to obtain the tree property simultaneously at large intervals of regular cardinals. The ultimate goal being to force the tree property at every regular cardinal greater than $\aleph_1$. By a theorem of Specker a positive answer would require many failures of SCH.
We show that from large cardinals, we can force the tree property at every regular cardinal in the interval $[\aleph_2, \aleph_{\omega^2}+2]$ with $\aleph_{\omega^2}$ a strong limit.
This is joint work with Cummings, Hayut, Magidor, Neeman, and Unger.
April 11, 2023
Maxwell Levin :
4:30 p.m. in 636 SEO
Abstract
Disjoint stationary sequences were introduced by Krueger to
answer a question about forcings that add clubs through stationary
sets.
We will discuss a version of Mitchell forcing that adds a disjoint
stationary sequence (given a sufficient large cardinal). The benefit of
this version is that it comes with an Abraham-style projection
analysis. This allows us to obtain disjoint stationary sequences on
successive cardinals, thus answering a couple of Krueger's questions.
Time permitting, we will discuss related issues.
Natasha Dobrinen :
3:30 p.m. in 636 SEO
Abstract
Generalizations of Ramsey's Theorem to colorings of infinite sets proceeds via topological considerations. The Galvin-Prikry Theorem states that Borel subsets of the Baire space are Ramsey. Silver extended this to analytic sets, and Ellentuck gave a topological characterization of Ramsey sets in terms of the property of Baire in the Vietoris topology.
We extend these theorems to several classes of countable homogeneous structures, answering a question of Kechris, Pestov, and Todorcevic. An obstruction to exact analogues of Galvin-Prikry or Ellentuck is the presence of big Ramsey degrees. We will discuss how different properties of the structures affect which analogues have been proved. Presented is work of the speaker for SDAP+ structures, and joint work with Zucker for binary finitely constrained FAP classes. In both works, the pigeonhole principle is achieved in ZFC by repeated applications of the forcing mechanism to find a finite object with desired properties. A feature of the work with Zucker is showing that we can weaken one of Todorcevic's four axioms guaranteeing a Ramsey space, and still achieve the same conclusion. These axioms are built on prior work of Carlson and Simpson developing topological Ramsey space theory.
April 18, 2023
Gabriel Conant :
4 p.m. in 636 SEO
Abstract
The starting point for this talk will be the recent discovery (initially made by Alex Kruckman) that a certain well-known basic axiom of model theoretic dividing is actually false. In the attempts to pick up the pieces around this situation, a number of new results and interesting examples have emerged, which I will discuss. One positive result from joint work with J. Hanson is a metric adaptation of PM Neumann’s Lemma (in the study of finite permutation group), which is used to prove ``full existence” for algebraic independence in continuous logic, answering a question of Andrews, Goldbring, and Keisler. More recently, in joint work with Kruckman, we have found a number of strange counterexamples demonstrating peculiar behavior of dividing in non-simple theories. In particular these examples give negative answers to a question of Adler (about the relationship between forking and dividing in theories where all sets are extension bases) and a question of Kaplan and Ramsey (about the relationship between forking and dividing in NSOP1 theories).
April 25, 2023
Alexi Block-Gorman :
4 p.m. in 636 SEO
Abstract
Büchi automata are the natural extension of finite automata to a model of computation that accepts infinite-length inputs. We say a subset X of the reals is k-regular if there is a Büchi automaton that accepts (one of) the base-k representations of every element of X, and rejects the base-k representations of each element in its complement. These sets often exhibit fractal-like behavior--e.g., the Cantor set is 3-regular. Let V_k be a ternary predicate such that V_k(x,u,d) holds if and only if u is an integer power of k and d is the coefficient of the term u in some base-k expansion of x. For a fixed k and for each natural number n, all of the k-regular subsets of Euclidean space definable in the expansion of the ordered additive group of reals by the predicate V_k. In this talk, we will discuss the significance of the ordered additive group of reals by V_k (and its reducts) from the perspectives of tame geometry and neostability. We will also discuss current and ongoing progress toward a characterization of the reducts of this structure in terms of definability, neostability, and fractal dimensions.
Aug. 29, 2023
Matthew Harrison-Trainor :
4 p.m. in 636 SEO
Abstract
Given a structure $\mathcal{A}$, we can form the back-and-forth tree $T(\mathcal{A})$. The nodes of this tree are the finite tuples from $\mathcal{A}$, ordered by extension. Each node is labeled by its atomic type. The standard back-and-forth argument shows that the tree of tuples is a complete isomorphism invariant, and captures the full theory of the structure in infinitary logic. The tree of tuples was also used to show the Borel-completeness of the class of linear orders. However one cannot compute back a copy of the original structure from the tree. We will talk about this result as well as its consequences.
Sept. 5, 2023
Ronnie Nagloo :
4 p.m. in 636 SEO
Sept. 12, 2023
Scott Mutchnik :
4 p.m. in 636 SEO
Abstract
Since at least the first decade of the 21st century, two questions have troubled model theorists: how can we extend (neo)stability-theoretic methods beyond simplicity (and now, beyond NSOP_2), and is NSOP_2 equal to NSOP_3? Recent progress on the equality of NSOP_1 and NSOP_2 has offered hope that these two questions are related. However, while new stability-theoretic relations such as Conant-independence and n-ð-independence have proven promising in making sense of the higher NSOP_n hierarchy, positive global consequences of NSOP_n, for n > 2, have continued to elude us. We discuss our recent finding of the first such results, on NSOP_3. While it is still open whether NSOP_3 coincides with NSOP_2 (i.e. with NSOP_1), these results are concrete in that they do not, in general, hold in NSOP_4 theories. Previously, NSOP_3 has been thought of as very different from NTP_2, in the sense that there is no known NSOP_3 NTP_2 theory which is not also simple. It is therefore surprising that our results on NSOP_3 theories illustrate similar behavior to NTP_2 theories.
Sept. 19, 2023
Maryanthe Malliaris :
4 p.m. in 636 SEO
Abstract
The talk will be about shearing, which is like dividing (see arXiv:2109.12642) in certain ways.
Sept. 26, 2023
John Baldwin :
4 p.m. in 636 SEO
Abstract
Let $M$ be strongly minimal and constructed by a `Hrushovski
construction' with a single ternary relation. If the Hrushovski
algebraization function $\mu$ is in a certain class $\Tscr$ ($\mu$ triples)
we show that for independent $I$ with $|I| >1$, $\dcl^*(I)= \emptyset$ (*
means not in $\dcl$ of a proper subset). This implies the only definable
truly $n$-ary functions $f$ ($f$ `depends' on each argument), occur when
$n=1$.
We prove
% , indicating the dependence on $\mu$,
for Hrushovski's original construction and for the strongly minimal
$k$-Steiner systems of Baldwin and Paolini that the symmetric definable
closure, $\sdcl^*(I) =\emptyset$ (Definition~\ref{defsdcl}). Thus, no such
theory admits elimination of imaginaries. As, we show that in an arbitrary
strongly minimal theory, elimination of imaginaries implies $\sdcl^*(I)
\neq \emptyset$.
Oct. 3, 2023
Atticus Stonestrom :
4 p.m. in 636 SEO
Abstract
Dp-minimality is a kind of abstract model-theoretic "one-dimensionality" condition, satisfied for example by superstable theories of U-rank 1 and o-minimal theories. In this talk we will introduce dp-minimality, and then discuss some results on dp-minimal groups: namely, every torsion-free dp-minimal group is abelian, every stable dp-minimal group is solvable-by-finite, and every distal dp-minimal group is nilpotent-by-finite.
Oct. 6, 2023
John Alexander Cruz :
noon in the internet
Abstract
In this series of talks I will introduce some Schwarzian differential equations that can be obtained from mirror maps for Calabi-Yau manifolds, in particular, hypersurfaces in projective spaces. I will propose a set of expectations about the model theoretic properties of those equations that should extend properties studied in the works of Aslanyan, Freitag, Nagloo, Scanlon and others. The goal is to formulate those expectations as theorems and discuss the strategy of their proof. At the end, I will mention some possible applications of the model theoretic properties in the study of mirror symmetry.
Oct. 10, 2023
Don Stull :
4 p.m. in 636 SEO
Abstract
Recent work has shown that techniques from computability theory and algorithmic randomness can be used to understand questions in classical geometric measure theory. One of the central problems in geometric measure theory is Falconer's distance set problem. Give a set E in the plane, and a point x, the pinned distance set of E with respect to x is the set of distances between x and the points in E. In this talk, we will discuss ongoing progress on this problem, and present improved lower bounds for both the Hausdorff and packing dimensions of pinned distance sets. We also discuss the computability-theoretic methods used to achieve these bounds.
Oct. 17, 2023
James Freitag :
4 p.m. in 636 SEO
Abstract
One of the amazing things about Galois theory is that sometimes you can convert a natural motivational question to a question about group actions. This is incredibly useful, since group theory has some of the deepest classification results in mathematics.
In model theory, we have a definable version of Galois theory, and numerous questions can be reduced to the basic building blocks in our category, definably primitive group actions. We'll talk about this general picture and some natural open questions.
Oct. 24, 2023
Matthew Harrison-Trainor :
4 p.m. in 636 SEO
Abstract
I will talk about an Effective Gelfand Duality between a compact topological space X and its algebra of functions. We can use that to reduce the homeomorphism problem for compact topological spaces to the isomorphism problem for C*-algebras, which is known to be Borel. One can then ask, for a particular space X, what is the complexity of the set of homeomorphic copies of X?
Nov. 21, 2023
Will Adkisson :
4 p.m. in 636 SEO
Abstract
Stationary sets are a fundamental concept in set theory, but their definition only makes sense at regular cardinals. We will describe mutual stationarity, a property that can be viewed as an analog of stationarity for singular cardinals, and discuss how it interacts with other combinatorial properties at or near $\aleph_\omega$. In particular, we will discuss stationary reflection and the failure of the Singular Cardinal Hypothesis.
Jan. 23, 2024
Aaron Anderson :
11 a.m. in 636 SEO
Abstract
We examine distal theories and structures in the context of continuous logic, providing several equivalent definitions.
By studying the combinatorics of fuzzy VC-classes, we find continuous versions of (strong) honest definitions and distal cell decompositions.
By studying generically stable Keisler measures in continuous logic, we apply the theory of continuous distality to analytic versions of graph regularity.
We will also present some examples of distal metric structures, including dual linear continua and a continuous version of o-minimality.
Feb. 16, 2024
Ronnie Nagloo :
noon in the internet
Abstract
The Schwarzian equations appear as the uniformizing
differential equations for Fuchsian covering maps
(or uniformizers). In this talk I will survey the work
with Blázquez-Sanz, Casale and Freitag, around
using model theoretic techniques to study these equations.
I will also try and discuss some of the questions asked about them
in the previous talks.
Feb. 21, 2024
James Freitag :
4 p.m. in 712 SEO
Abstract
In the talk, we will explain what are the prospects for classifying which complex analytic functions are Pfaffian (in the sense of Khovanskii) and why you should care.
Feb. 28, 2024
Scott Mutchnik :
4 p.m. in 712 SEO
Abstract
Koponen has conjectured that all simple theories, with quantifier elimination in a finite relational language, are supersimple of finite rank (2016). In particular, she asks whether they are one-based (2014). We discuss our proof of this conjecture, which by Tomašić and Wagner’s results on pseudolinearity (2003), also implies one-basedness. Particularly, we prove that simplicity implies supersimplicity in this setting, thereby highlighting what Kennedy (2020) calls "the fragility of the syntax-semantics distinction.”
This is on joint work with John Baldwin and James Freitag.
March 6, 2024
Scott Mutchnik :
4 p.m. in 712 SEO
Abstract
We continue the proof of the first implication of the Koponen conjecture, showing that countably categorical n-ary simple theories are supersimple. Then we give our own proof, based on arguments of Palacín, that countably categorical, n-ary supersimple theories must have finite rank, using a quantity, F_Mb, that is in a certain sense dual to Freitag and Moosa’s degree of nonminimality. We also explain Tomašić and Wagner’s result on pseudolinearity, which depending on the strength used, either proves the implication from finite rank to one-based for omega-categorical n-ary theories, or proves the full implication from supersimple to one-based. If time permits, we explain the connection to some open questions on F_Mb.
This is on joint work with John Baldwin and James Freitag.
March 13, 2024
Matthew Harrison-Trainor :
4 p.m. in 712 SEO
Abstract
Given a set A, we say that A is introreducible if all subsets of A can compute A. I will talk about several results about introreducibility and the related notion where all subsets of A can compute some other set C.
March 27, 2024
Matthew Harrison-Trainor :
4 p.m. in 712 SEO
Abstract
Given a set A, we say that A is introreducible if all subsets of A can compute A. I will continue by talking about more results on introreducibility, particularly touching on uniformity and the difference between computation and enumeration.
April 3, 2024
John Baldwin :
4 p.m. in 712 SEO
Abstract
We survey variants of the Hrushovki non-locally modular strongly minimal
sets construction that get more combinatorial examples and show the basic
construction is essentially unary and thus does not eliminate imaginaries.
A $t-(\kappa,k,s)$ block design is a set of $\kappa$ elements and a
collection of $k$-element subsets $B$ of $P$ (called blocks) with the
property that each $t$-element subset of $P$ occurs in exactly $s$ blocks.
A $k$-Steiner system is a $2-(\kappa,k,1)$ system.
Using variants of the Hrushovski method we construct infinite block
designs and Steiner systems that are a) $\aleph_1$-categorical and with
more work b) have $t$-transitive automorphism groups for prescribed $t$.
The high transivity is on-going work with Freitag and Mutchnik.
The strongly minimal Steiner $k$-Steiner system $(M,R)$ from
Baldwin and Paolini can be `coordinatized' in the sense of Ganter and Werner by
a quasigroup if $k$ is a prime-power. But for the basic construction this
coordinatization is never definable in $(M,R)$. In
almost all cases the theory does not admit elimination of imaginaries. Nevertheless, by refining the construction, if $k$ is a
prime power there is a $(2,k)$-variety of quasigroups which is strongly
minimal and definably coordinatizes a Steiner $k$-system.
April 10, 2024
Kevin Zhou :
4 p.m. in 712 SEO
Abstract
Hereditary properties of graphs, which are classes of finite graphs closed under isomorphism and induced subgraph, are well-studied objects in combinatorics. Of particular interest are the possible asymptotic growth rates of the number of graphs with vertex set [n] in a hereditary property. In the 2000’s, a series of papers gave a classification into only four discrete growth rates, which was later generalized by Laskowski and Terry to hereditary properties in any finite relational language. The problem is of particular model-theoretic interest because hereditary properties are precisely the classes of models of universal theories.
A closely related problem related to statistical relational learning is that of weighted model counting, in which weights are assigned to each relation in the language and the goal is to efficiently compute the weighted count of the models of a given sentence. The advantage of working in the weighted setting is the existence of a Skolemization procedure - for any weighted model counting problem, there is an equivalent problem where the sentence is universal, while still staying in the finite relational setting. I will discuss some new connections between the work on classifying the growth rates of hereditary properties and efficient weighted model counting.
April 17, 2024
Patrick Lutz :
4 p.m. in 712 SEO
Abstract
The study of the structure of countable Borel equivalence relations under Borel reducibility has been a major focus of descriptive set theory over the past few decades. However, many open questions remain, many of which involve the hyperfinite equivalence relations (essentially the simplest nontrivial countable Borel equivalence relations). In order to better understand these questions, Conley and Miller introduced a weakening of Borel reducibility, known as measure reducibility. They then answered the analogues for measure reducibility of several of these questions. However, they left open at least one such question. Namely, is there a minimal (in the sense of measure reducibility) non-hyperfinite equivalence relation? Such an object is called a "measure successor of E_0." In ongoing work, Jan Grebik and I have isolated a combinatorial property of group actions on Polish spaces which implies that the associated orbit equivalence relation is a measure successor of E_0 and found several examples of group actions which are plausible candidates for satisfying this condition. The combinatorial property we have identified is a strong form of expansion which we call "lossless expansion" after a similar property studied in computer science and combinatorics. I will explain the context for Conley and Miller's question and the combinatorial condition that Grebik and I have isolated.
April 24, 2024
James Walsh :
4 p.m. in 712 SEO
Abstract
We present an analogue of Gödel’s second incompleteness theorem. Whereas Gödel showed that sufficiently strong theories that are $\Pi^0_1$-sound and $\Sigma^0_1$-definable do not prove their own $\Pi^0_1$-soundness, we prove that sufficiently strong theories that are $\Pi^1_1$-sound and $\Sigma^1_1$-definable do not prove their own $\Pi^1_1$-soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.
We will then turn to characterizations of ordinal analysis itself. One of the main goals of ordinal analysis is measuring the “strength” of theories by calculating their proof-theoretic ordinals. But in what sense do proof-theoretic ordinals measure the strength of theories? What is the attendant notion of strength? We provide some abstract answers to this question.
Sept. 10, 2024
Caroline Terry :
4 p.m. in 636 SEO
Abstract
Many tools have been developed in combinatorics to study global structure in finite graphs. One such tool is called Szemer'{e}di’s regularity lemma, which gives a structural decomposition for any large finite graph. Beginning with work of Alon-Fischer-Newman, Lov\'{a}sz-Szegedy, and Malliaris-Shelah, it has been shown over the last 15 years that regularity lemmas can be used to detect structural dichotomies in graphs, and that these dichotomies have deep connections to model theory. One striking example is a dichotomy in the size of regular partitions, first observed by Alon-Fox-Zhao. Specifically, if a hereditary graph property $\mathcal{H}$ has finite VC-dimension, then results of Alon-Fischer-Newman and Lovász-Szegedy imply all graphs in $\mathcal{H}$ have regular partitions of size polynomial is $1/\epsilon$. On the other hand, if $\mathcal{H}$ has infinite VC-dimension, then results of Gowers and Fox-Lov\'{a}sz show there are graphs in $\mathcal{H}$ whose smallest $1/\epsilon$-regular partition has size at least an exponential tower of height polynomial in $1/\epsilon$. In this talk, I present several analogous dichotomies in the setting of hereditary properties of 3-uniform hypergraphs.
Sept. 17, 2024
James Freitag :
4 p.m. in 636 SEO
Abstract
Controlling various aspects of forking is an important ingredient in many settings of both pure and applied model theory. In this talk, I'll talk about how to make some of these techniques effective or quantitative.
Sept. 24, 2024
Alberto Miguel Gómez :
4 p.m. in 636 SEO
Abstract
3-hypertournaments are combinatorial structures that generalize tournaments to the ternary relational case in a similar way that 3-hypergraphs generalize graphs. Recently, Cherlin, Hubička, Konečny, and Nešetřil have identified a countable homogeneous 3-hypertournament with some wild behavior from the structural-Ramsey-theoretic point of view. In this talk, I will show that this behavior has a model-theoretic counterpart: namely, its theory is strictly NSOP4. Furthermore, the usual criteria from the literature do not apply in this case, making this a novel example of an NSOP4 theory. In this talk, I will discuss the proof of this fact and how it relates to the other known examples of NSOP4 in light of recent developments in the area.
Sept. 27, 2024
David Marker :
10 a.m. in 304 Taft Hall
Abstract
Shelah showed that it is consistent that there are uncountable rigid non-archimedean real closed fields and, later, he and Mekler proved this in ZFC. Answering a question of Enayat, Charlie Steinhorn and I show that there are countable rigid non-archimedean real closed fields by constructing one of transcendence degree two.
Oct. 1, 2024
Gabriela Laboska :
4 p.m. in 636 SEO
Abstract
An inhomogeneous system of linear equations over a ring $R$ is partition
regular if for any finite coloring of $R$, the system has a monochromatic
solution. In 1933, Rado showed that an inhomogeneous system is partition
regular over $\mathbb{Z}$ if and only if it has a constant solution.
Following a similar approach, Byszewski and Krawczyk showed that the
result holds over any integral domain. In 2020, Leader and Russell
generalized this over any commutative ring $R$, with a more direct
proof than what was previously used. We analyze some of these combinatorial
results from a computability-theoretic point of view, starting with
a theorem by Straus used directly or as a motivation to many of the
previous results on the subject.
Oct. 15, 2024
Ben Castle :
4 p.m. in 636 SEO
Abstract
Many authors have asked whether the Zilber trichotomy holds for strongly minimal structures definable (resp. interpretable) in structures of a particular class, say $\mathcal C$ (examples include $\mathcal C$ = all algebraically closed fields, algebraically closed valued fields, and o-minimal structures).
Typically the "interpretable" case is seen as harder than the "definable" case (often significantly so). On the other hand, recent work with Hasson and Ye suggests otherwise. In particular, the following all turn out to be formal implications:
1. For $\mathcal C$ = algebraically closed fields of residue characteristic zero, the definable case implies the interpretable case.
2. For $\mathcal C$ = real closed valued fields, the definable case implies the interpretable case.
3. The definable case for $\mathcal C$ = o-minimal fields implies the interpretable case for $\mathcal C$ = all o-minimal structures.
These results all use variants of the same method, which (by the standards of trichotomy proofs) is surprisingly short and accessible. In this talk, I will attempt to introduce this method by giving a complete proof of (3) (which is the easiest), and discussing how to adapt the method to (1) and (2).
Oct. 22, 2024
Hongyu Zhu :
4 p.m. in 636 SEO
Abstract
Viewed as a subset of Cantor space, the class of countable models Mod(T) of any first order theory T is always Borel. A natural question, then, is the relationship between its descriptive complexity and the complexity of the underlying theory. Using theorems of López-Escobar and Solovay, we give a more precise characterization of the complexity of Mod(T) in terms of that of T. We also discuss some applications to models of PA and infinitary logic. (This is based on joint work with Andrews, Gonzalez, Lempp, Rossegger, and related to recent work of Enayat and Visser.)
Oct. 29, 2024
Ioannis Eleftheriadis :
4 p.m. in 636 SEO
Abstract
Abstract: Algorithmic meta-theorems are uniform results in complexity theory that simultaneously yield an efficient solution to a wide class of computational problems, typically those expressible by a sentence of a certain logic. In the context of graphs, various research programmes have identified combinatorial conditions that provide dividing lines for the existence of uniform algorithmic results, such as the graph minors theory of Robertson and Seymour, the sparsity theory of Nesetril and Ossona de Mendez, and the twin-width theory of Bonnet et al. Recent years have seen the emergence of model-theoretic techniques in unifying these programmes and understanding the most general conditions that effectuate algorithmic tractability. This talk will survey the structural tractability programme, and present recent progress towards a conjecture that places monadic NIP as the limit to algorithmic tractability for first-order expressible problems. This is based on joint work with Jan Dreier, Nikolas Mahlmann, Rose McCarty, Michal Pilipczuk, and Szymon Torunczyk (FOCS 2024).
Nov. 5, 2024
Leonardo Coregliano :
4 p.m. in 636 SEO
Abstract
A random structure on a vertex set $V$ (in a fixed finite relational language) is exchangeable if
its distribution is invariant under permutations of $V$. The Aldous--Hoover Theorem says all such
distributions are generated from a collection of i.i.d. variables on $[0,1]$, one for each subset
of $V$, using a simple rule that was later called "Euclidean structure" by combinatorialists. As the
name suggests, an Euclidean structure resembles a relational structure over $[0,1]$, except for the
presence of "higher-order variables".
One of the original questions of Hoover was to determine which such distributions admit simpler
descriptions that do not depend on certain variables. Very little progress was obtained in this
problem until it got revisited under the light of the theories of limits of combinatorial objects
and quasirandomness. It turns out that asking for a representation of an exchangeable hypergraph in
which the Euclidean structure is a usual (measurable) relational structure over $[0,1]$ (i.e., which
does not need any higher-order variables) is equivalent to asking for "tamer" Szemerédi regularity
lemmas and was solved using the theory of hypergraphons.
The dual problem of determining when there is a representation that does not need any low-order
variable is more closely related to quasirandomness, which informally is the property of "lack of
correlation with simple structures".
In this talk, I will introduce exchangeability and quasirandomness theory and talk about recent
progress on the aforementioned dual problem. I will assume familiarity with basic logic/model
theory, but no prior knowledge in extremal combinatorics, limit theory or quasirandomness will be
necessary.
This talk is based on joint works with Alexander Razborov and Henry Towsner.
Bruno Poizat :
2 p.m. in 636 SEO
Abstract
We study the connections between the automorphisms of an AC field K and the automorphisms of a structure S definable in it. Our methods are model theoretic and use a minimum of algebra. As an application, we show that the automorphisms of a simple algebraic group G over K acts on a copy L of K, with kernel the geometric automorphisms of G.
Nov. 13, 2024
Leo Jimenez :
10 a.m. in 304 Taft
Abstract
When solving a differential equation, one sometimes finds that solutions can be expressed using a finite number of fixed, particular solutions, and some complex numbers. As an example, the set of solutions of a linear differential equation is a finite-dimensional complex vector space. A model-theoretic incarnation of this phenomenon is internality to the constants in a differentially closed field of characteristic zero. In this talk, I will define what this means, and discuss some recent progress, joint with Christine Eagles, on finding methods to determine whether the solution set of a differential equation is internal. As a corollary, we obtain a criterion for solutions to be orthogonal to the constants, and in particular not Liouvillian. I will show a concrete application to Lotka-Volterra systems.
Nov. 19, 2024
John Baldwin :
4 p.m. in 636 SEO
Abstract
We sketch the proof that if a Fuchsian group $\Gamma$ has finite index in its commensurator (i.e. is non-arithmetical), then the
theory of its two sorted universal cover structure is axiomatized by a sentence in $L_{\omega_1,\omega}$ that
is categorical in uncountable powers. This is a simpler version of the earlier result in the arithmetic
case (Daw & Harris; Eterović). The argument is structured rather differently and clarifies the roles of model theory and
geometry in the earlier case by replacing some `geometric/number theoretic' arguments by model theoretic ones. We point to the
role of finite index in allowing these simplifications. Joint work with Joel Nagloo and incorporating some earlier work
with Andres Villaveces in a survey of Zilber's `logically perfect structures are $L_{\omega_1,\omega}(Q)$-categorical in
uncountable power.
Nov. 26, 2024
Gabriel Conant :
4 p.m. in 636 SEO
Abstract
In 1954, Følner proved the following discrete analogue of Steinhaus's Theorem: If $A$ is a set of positive upper Banach density in an abelian group $G$, then $A-A$ almost (i.e., modulo zero Banach density) contains a neighborhood of the identity in the Bohr topology on $G$. An analogous result for countable discrete amenable groups was proved by Beiglbock, Bergelson, and Fish in 2010 using ergodic theory. In this talk, I will present a proof of this result which is valid for arbitrary discrete amenable groups and is directly inspired by fundamental ideas and facts from local stability theory in continuous logic. Familiarity with continuous logic (or any logic) will not be necessary.
Dec. 3, 2024
Rachel Greenfeld :
4 p.m. in 636 SEO
Abstract
Translational tiling is a covering of a space (such as Euclidean space) using translated copies of one building block, called a "translational tile," without any positive measure overlaps. Can we determine whether a given set is a translational tile? Does any translational tile admit a periodic tiling? A well-known argument shows that these two questions are closely related. In the talk, we will discuss this relation and present recent developments, joint with Terence Tao, establishing answers to both questions.
Jan. 23, 2025
Christian d'Elbée :
3:30 p.m. in 636 SEO
Abstract
Recall that a structure (group, Lie algebra, associative algebra, etc) M is omega-categorical if there is a unique countable model of its first-order theory, up to isomorphism. This model theoretic notion has a dynamical definition: M is omega-categorical if and only if there are only finitely many orbits in the component-wise action of Aut(M) on the cartesian power M^n, for all natural number n.
In 1981, Wilson conjectured that any omega-categorical locally nilpotent group is nilpotent. If true, a quite satisfactory decomposition of omega-categorical groups would follow. This conjecture is very much open more than 40 years later. The analogue statement for Lie algebras (every locally nilpotent omega-categorical Lie algebra is nilpotent) is also open and, as it turns out, it reduces to proving that for each n and prime p, every omega-categorical n-Engel Lie algebra over F_p is nilpotent. As for associative algebras, the analogous question was already answered by Cherlin in 1980: every locally nilpotent omega-categorical ring is nilpotent. We see the Wilson conjecture for Lie algebra as a bridge between the result of Cherlin and the original question of Wilson for omega-categorical groups.
The question of Wilson, for groups, for Lie algebras or for associative algebras are connected to classical nilpotency problems such as the Burnside problem, the Kurosh problem or the problem of local nilpotency of n-Engel groups.
Using a classical result of Zelmanov, the Wilson conjecture for omega-categorical Lie algebras is true asymptotically in the following sense: for each n, every n-Engel Lie algebra over F_p is nilpotent for all but finitely many p's. The situation for small values of the pair (n,p) is as follows:
. Every 2-Engel Lie algebra is nilpotent (Higgins 1954),
. Every 3-Engel Lie algebra over F_p with p\neq 2,5 is nilpotent (Higgins 1954),
. Every 4-Engel Lie algebra over F_p is nilpotent for p\neq 2,3,5. (Higgins 1954, Kostrikin 1959),
. Every 5-Engel Lie algebra over F_p is nilpotent for p\neq 2,3,5,7 (Vaughan-Lee, 2024).
In other words, for (n,p) = (3,2), (3,5), (4,2), (4,3), (4,5),... It is known that n-Engel Lie algebras of char p are not globally nilpotent. Our goal, on the long run, is to prove that for those values of (n,p), omega-categorical n-Engel Lie algebra of characteristic p are nilpotent.
We have recently dealt with the cases (n,p) = (3,5) and (n,p) = (4,3), and the proofs are different both in taste and method.
The goal of the talk is to present a proof that every omega-categorical 4-Engel Lie algebras of characteristic 3 is nilpotent. Our solution of the case at hand consists in adapting in the definable context some classical tools for studying Engel Lie algebras, appearing earlier in the work of Higgins, Kostrikin, Zelmanov, Vaughan-Lee, Traustason and others. Our solution involves the use of computer algebra.
Feb. 4, 2025
Kyle Gannon :
3:30 p.m. in 636 SEO
Abstract
This talk is motivated by the following two soft questions: How do we sample an
infinite sequence from a first order structure? What model theoretic properties might hold on
almost all sampled sequences? We advance a plausible framework in an attempt to answer these
kinds of questions. The central object of this talk is a probability space. The underlying set of
our space is a standard model theoretic object, namely the space of types in countably many
variables over a monster model. Our probability measure is an iterated Morley product of a
fixed Borel-definable Keisler measure. Choosing a point randomly in this space with respect to
our distribution yields a random generic type in infinitely many variables. We are interested in
which model theoretic events hold for almost all random generic types. Two different kinds of
events will be discussed: (1) The event that the induced structure on a random generic type is
isomorphic to a fixed structure; (2) the event that a random generic type witnesses a dividing
line.
Feb. 11, 2025
James Freitag :
3:30 p.m. in 636 SEO
Abstract
This is a mostly expository talk about several problems for algebraic dynamics (algebraic varieties with rational self-maps). We will introduce a recent construction of Kamensky and Moosa yielding a binding group for isotrivial algebraic dynamics. We will talk about applications of understanding this group action including the Zariski-dense orbit conjecture, the Dixmier-Moeglin conjecture, and the existence of invariant rational factors.
Feb. 18, 2025
Matthew Harrison-Trainor :
3:30 p.m. in 636 SEO
Abstract
This talk will be an introductory talk aimed at graduate students. I will introduce infinitary logic, the Scott analysis and Scott sentences, and Montalban's robust Scott rank. We will give some proofs and proof sketches.
Feb. 25, 2025
John Baldwin :
3:30 p.m. in 636 SEO
Abstract
My approach is historical, beginning with Morley's theorem, and continuing with the connections of $L_{\omega_1,\omega}$
with atomic models of a first order theory.
I discuss work Laskowsi, Shelah, and I have been working on since at
least 2013 on whether an $L_{\omega_1,\omega}$ class $K$ in a countable vocabularyhas
unboundedly large models and/or is stable in $\aleph_0$.
A first order
theory is categorical in $\aleph_1$ iff it is $\omega$-stable and has no
two cardinal models; this characterization is easily seen to be absolute.
Already in [Sh87] Shelah had given an example of an
$L_{\omega_1,\omega}(Q)$ sentence that is categorical under MA but not
under $2^{\aleph_0}<2^{\aleph_1}$. Whether there is similar example for
$L_{\omega_1,\omega}$ remains open.
%The recent work goes in the other direction, trying to find tractable absolute conditions equivalent to $\aleph_1$-categoricity.
In \cite{BLS16}, we reformulated the problem as the study of atomic models of first order
theories and introduced the notions of
pseudo-algebracity and of a pseudo-minimal set as an analog for
strong minimality. We proved: If a countable first order theory $T$ has
an atomic model and fewer than $2^{\aleph_1}$ models in $\aleph_1$ then
the pseudo-minimal types are dense.
In [BLS24] we showed that a sentence of $L_{\omega_1,\omega}$
that is categorical in $\aleph_0$ and $\aleph_1$ and has a model in
$\beth_1^+$ is $\omega$-stable. I will expound this line of work and the role of
the essential set-theoretic argument forcing argument (within ZFC). The most recent development
is an analog of $U$-rank in this setting.
March 4, 2025
Ronnie Chen :
3:30 p.m. in 636 SEO
Abstract
A standard technique in countable model theory is to define a topological space "continuously parametrizing" countable structures in some language, and then apply methods of descriptive set theory and topological dynamics to model-theoretic questions. In fact several versions of such a "space of models" are known, all of which are instances of the notion of an étale bundle (aka sheaf) of structures. We will give an introduction to the theory of étale bundles, and survey some of the countable model-theoretic machinery that may be naturally generalized to this context, including the Lopez-Escobar theorem, characterizations of Scott rank, and groupoid representations.
March 18, 2025
Christine Eagles :
3:30 p.m. in 636 SEO
Abstract
It is well known that in stable theories, we can understand finite dimensional types in terms of minimal types. We will talk about one such method which we call a composition analysis. We explore a uniqueness condition for a set of minimal types we associate to a type through the composition analysis. This is based on current work in progress.
April 1, 2025
Denis Hirschfeldt :
2:30 p.m. in 636 SEO
Abstract
The upper density of the symmetric difference between two sets of natural numbers gives a notion of distance that can be used to define a metric on the Turing degrees. By work of Monin, this metric is (0,1/2,1)-valued. A degree a is attractive if almost every degree is at distance 1/2 from a, and dispersive otherwise. I will discuss joint work with Jockusch and Schupp, as well as more recent work of Royer, on the distribution of attractive and dispersive degrees, and their connections with the interplay between effective randomness and genericity.
Andrew Marks :
4 p.m. in 636 SEO
Abstract
We discuss a technique called recursive compression for proving incomputability results. The method has developed independently in mathematics and theoretical computer science, and gives a way of showing some set is incomputable by reducing the halting problem to it. Recursive compression was used by Durand, Romashchenko, and Chen in 2008 to give a new proof that the Wang tiling problem is incomputable. In quantum information theory, recursive compression was used by Ji, Natarajan, Vidick, Wright, and Yuen in 2020 to prove the MIP*=RE result that the halting problem is reducible to approximating the quantum value of a nonlocal game. Their result implies a negative answer to the longstanding Connes embedding problem in operator algebras.
We formulate a general recursive compression lemma which abstracts the technique used in these applications. A recursive compression f of a set A ⊂ 2ω is a polytime computable function which takes as input a program e computing a string x in exponential time, and outputs a program f(e) computing a string y in polynomial time so that x ∈ A iff y ∈ A. If A has a recursive compression, and A and its complement are nonempty, then A is incomputable. We also show a converse of the recursive compression lemma: the halting problem is polytime reducible to an r.e. set if and only if there is a recursive compression. Finally, we generalize the recursive compression lemma throughout the arithmetical hierarchy, giving a way to show that a language is Σ0n-hard using recursive compression. This is joint work with Seyed Sajjad Nezhadi and Henry Yuen.
Natasha Dobrinen :
1 p.m. in 636 SEO
Abstract
In this talk, we will give an introduction to big Ramsey degrees of homogeneous structures. Then we will concentrate on computability theoretic aspects and discuss work done in the area by various logicians, including work by Cholak, Dobrinen, McCoy.
April 8, 2025
Nick Ramsey :
3:30 p.m. in 636 SEO
Abstract
A primitive permutation group (X,G) is a group G together with an action of G on X such that there are no nontrivial equivalence relations on X preserved by G. An rough classification of primitive permutation groups of finite Morley rank, modeled on the O'Nan-Scott theorem for finite primitive permutation groups, has been carried out by Macpherson and Pillay and this classification was then used by Borovik and Cherlin to prove that if (X,G) is a primitive permutation group of finite Morley rank, the rank of G can be bounded in terms of the rank of X. We study the analogous situation for pseudo-finite primitive permutation groups of finite SU-rank, building both on supersimple group theory and classification results of Liebeck, Macpherson, and Tent. This is joint work with Ulla Karhumäki.
April 15, 2025
Anand Pillay :
3:30 p.m. in 636 SEO
Abstract
We prove an "arithmetic" version of Tao's algebraic regularity lemma about graphs uniformly definable in finite fields. Namely with uniformly
definable pairs $(G,A)$ ($G$ group, $A$ subset) in place of a graph. We make connections with Green's arithmetic regularity lemma for finite dimensional vector spaces over $F_p$, and
results of Gowers on quasirandom groups. (Joint with Atticus Stonestrom.)
April 22, 2025
James Hanson :
3:30 p.m. in 636 SEO
Abstract
We will discuss a characterization of first-order theories realizing a certain combinatorial tree configuration in terms of special coheirs.
April 29, 2025
Nigel Pynn-Coates :
3:30 p.m. in 636 SEO
Abstract
The theory of closed H-fields is model complete and axiomatizes the
theory of transseries and maximal Hardy fields, as Aschenbrenner, Van
den Dries, and Van der Hoeven have shown in a long series of works. To
better understand large closed H-fields, such as maximal Hardy fields, I
recently extended this model completeness to the theory of tame pairs of
closed H-fields. Building on this work, I will explain how to extend
differential-algebraic dimension on a closed H-field to tame pairs of
closed H-fields so that it is a fibred dimension function in the sense
of [L. van den Dries, "Dimension of definable sets, algebraic
boundedness and Henselian fields", Ann. Pure Appl. Logic 45.2 (1989),
189–209] and the nonempty dimension zero definable sets are exactly the
nonempty discrete definable sets. The model-theoretic notion of
coanalyzability will also make an appearance.
Aug. 26, 2025
Scott Mutchnik :
2 p.m. in 427 SEO
Abstract
We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories.
Graduate students are particularly encouraged to attend.
Guy Fowler :
3 p.m. in 636 SEO
Abstract
The Zilber--Pink conjecture is a major open conjecture in arithmetic geometry. In this talk, I will describe joint work with Vahagn Aslanyan and Sebastian Eterović in which we prove the conjecture for subvarieties of powers of a modular curve that satisfy a suitable geometric genericity condition. Our approach uses the model theory of differential fields and Ax--Schanuel for the j-function. If time permits, I will also discuss progress in making these results effective.
Sept. 2, 2025
Scott Mutchnik :
2 p.m. in 427 SEO
Abstract
We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories.
Graduate students are particularly encouraged to attend.
Our goal for this meeting will be to prove Kim’s lemma for dividing, the equivalence of forking and dividing, and the symmetry of forking-independence, all over models in simple theories.
Scott Mutchnik :
3 p.m. in 636 SEO
Abstract
It was observed very recently that the classical SOP_n hierarchy, a family of approximations of the strict order property which Shelah introduced with the goal of classifying non-simple first-order theories, extends to a hierarchy of properties SOP_r for r any real number at least 3. However, it remains open whether the real-valued NSOP_r hierarchy is distinct from the original integer-valued NSOP_n hierarchy. To make this question more tractable, we can ask it at the quantifier-free level, obtaining a real-valued quantity, of independent combinatorial interest, associated with any hereditary class of finite structures.
While it is also open whether this quantity can have non-integer values, we can show that, in the case of a hereditary class defined by finitely many omitted weak substructures, it is an integer. We will discuss, and aim to prove, this result.
Sept. 9, 2025
Gabriel Conant :
3 p.m. in 636 SEO
Abstract
I will sketch a new proof of the "NIP Arithmetic Regularity Lemma" (first proved in 2018 with Pillay and Terry). This is joint work with Terry.
Michael Lange :
2 p.m. in 427 SEO
Abstract
We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories.
Graduate students are particularly encouraged to attend.
We will aim to finish proving Kim’s lemma for dividing, the equivalence of forking and dividing, and the symmetry of forking-independence, all over models in simple theories.
Sept. 16, 2025
Scott Mutchnik :
2 p.m. in 427 SEO
Abstract
We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories. Graduate students are particularly encouraged to attend.
We will introduce the class of NSOP_1 theories, which contains the class of simple theories while allowing for additional complexity, and to which many of the independence phenomena originating from stable and simple theories extend in new ways. Specifically, Chernikov, Kaplan and Ramsey show that Kim-independence, a version of forking-independence "at a generic scale," behaves similarly in NSOP_1 theories to how forking-independence behaves in simple theories. We will define Kim-independence and give the main ideas of the proofs of some of Kaplan and Ramsey's results on NSOP_1 theories, including Kim's lemma for Kim-dividing, the equivalence of Kim-forking and Kim-dividing, and potentially the symmetry of Kim-independence.
Yutong Duan :
3 p.m. in 636 SEO
Abstract
A central objective in model theory of differentially closed field of characteristic zero is the classification of differential equations in terms of their geometric complexity. In this talk, I will present a new result on the Lotka–Volterra system. We show that, apart from a single exceptional choice of parameters, the system is strongly minimal, geometrically trivial and strictly disintegrated.
Sept. 23, 2025
Scott Mutchnik :
2 p.m. in 427 SEO
Abstract
We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories. Graduate students are particularly encouraged to attend.
In this session, we will introduce the stable forking conjecture, with the goal of recounting Brower's striking argument about forking with types of rank two.
Oct. 7, 2025
Scott Mutchnik :
2 p.m. in 427 SEO
Abstract
We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories.
Graduate students are particularly encouraged to attend.
We will show that the stability of the forking relation, a global variant of the stable forking conjecture, is equivalent for supersimple countably categorical theories to the stable forking conjecture. We then discuss the simple Kim-forking conjecture in NSOP_1 theories, and prove cases of this conjecture from joint work with Baldwin and Freitag, including an infinite-variable global variant in a general NSOP_1 theory, a finite-variable global variant in the case of finite F_Mb, and the full conclusion of the simple Kim-forking conjecture, given enough indices, for forking with realizations of an isolated type with the definable Morley property. All three of these results use a strong version of Kim’s lemma in NSOP_1 theories, due to Kaplan and Ramsey, which says that all Kim-independent Morley sequences exhibit Kim-dividing.
Oct. 14, 2025
Aris Papadopoulos :
3 p.m. in 636 SEO
Abstract
A shower thought that anyone interested in graph theory must have had at some point in their lives is the following: `How “sparse" must a given bipartite graph be, if I know that it has no “dense” subgraphs?’. This curiosity definitely crossed the mind of Polish mathematician K. Zarankiewicz, who asked a version of this question formally in 1951. In the years that followed, many central figures in the development of extremal combinatorics contemplated this problem, giving various kinds of answers. Some of these will be surveyed in the first part of my talk.
So far so good, but this is a logic seminar and the title says the words “Model Theory"… In the second part of my talk, I will discuss how the celebrated Szemerédi-Trotter theorem gave a starting point to the study of Zarankiewicz’s problem in “geometric” contexts, and how the language of model theory has been able to capture exactly what these contexts are. I will then ramble about improvements to the classical answers to Zarankiewicz’s problem, when we restrict our attention to semilinear/semibounded o-minimal structures, Presburger arithmetic, and various kinds of Hrushovski constructions.
The new results that will appear in the talk have been obtained jointly with Pantelis Eleftheriou.
Scott Mutchnik :
2 p.m. in 427 SEO
Abstract
We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories. Graduate students are particularly encouraged to attend.
Continuing from last time (when we motivated the shift from local to global versions of the stable forking conjecture), we will discuss the simple Kim-forking conjecture in NSOP_1 theories, and prove cases of this conjecture from joint work with Baldwin and Freitag, including an infinite-variable global variant in a general NSOP_1 theory, a finite-variable global variant in the case of finite F_Mb, and the full conclusion of the simple Kim-forking conjecture, given enough indices, for forking with realizations of an isolated type with the definable Morley property. All three of these results use a strong version of Kim’s lemma in NSOP_1 theories, due to Kaplan and Ramsey, which says that all Kim-independent Morley sequences exhibit Kim-dividing.
Oct. 21, 2025
Scott Mutchnik :
2 p.m. in 427 SEO
Abstract
We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories.
Graduate students are particularly encouraged to attend.
We will discuss more cases of the simple Kim-forking conjecture proven in joint work with John Baldwin and James Freitag, including a finite-variable global variant of the simple Kim-forking conjecture in the case of finite F_Mb, and the full conclusion of the simple Kim-forking conjecture, given enough indices, for forking with realizations of an isolated type with the definable Morley property. Time permitting, we will then discuss examples of finite F_Mb and the definable Morley property, which measure dependence within the indiscernible sequences in a type in quantitative and qualitative ways.
Artem Chernikov :
3 p.m. in 636 SEO
Abstract
We show that k-ary functions giving the measure of the intersection of multi-parametric families of sets in probability spaces, e.g. $(x,y,z)\in X\times Y\times Z\mapsto \mu(P_{x,y}\cap Q_{x,z}\cap R_{y,z})$, satisfy a particularly strong form of hypergraph regularity. More generally, this applies to the (integral) averages of continuous combinations of functions of smaller arity. This result is connected to higher arity stability in (continuous) model theory. In relation to that, we demonstrate that all 3-hypergraphs embedding both into the half-simplex and into $GS(\mathbb{F}_3)$, the two known sources of failure of ternary stability, do satisfy an analogous regularity lemma -- hence, unlike classical stability, strong ternary stability cannot be characterized simply by excluded hypergraphs.
Oct. 28, 2025
Scott Mutchnik :
2 p.m. in 427 SEO
Abstract
We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories.
Graduate students are particularly encouraged to attend.
We will prove the simple Kim-forking conjecture for isolated types with the definable Morley property, and then discuss examples of the definable Morley property and finite F_Mb(p).
Gabriel Day :
3 p.m. in 636 SEO
Nov. 4, 2025
Wei Li :
3 p.m. in 636 SEO
Abstract
We analyze the behavior of systems of algebraic differential equations when considered as systems of difference-differential equations, with special emphasis on systems which define strongly minimal sets relative to the theory $DCF_{0,n}$ of differentially closed fields of characteristic zero with $n$ distinguished commuting derivations. We show that if $X$ is a strongly minimal set relative to $DCF_{0,n}$ defined by a finite system of algebraic partial differential equations and the forking geometry on $X$ is geometrically trivial, then $X$ remains minimal when regarded as definable set relative to the theory $DCFA_{0,n}$ of difference-differentially closed fields of characteristic zero with $n$ commuting derivations. We illustrate this theorem by describing in detail the possible difference-differential equations consistent with differential equations of the form $y' = f(y)$ for cubic polynomial $f$ over constants.
Scott Mutchnik :
2 p.m. in 427 SEO
Abstract
We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories.
Graduate students are particularly encouraged to attend.
We will discuss examples of, and connections between, the definable Morley property and finite F_Mb, concentrating particularly on the case of the generic projective planes of Conant and Kruckman. We initially motivated these properties using the simple Kim-forking conjecture, and if there is enough time, we will begin discussing these properties' connection with pseudolinearity, which offers another motivation.
Nov. 11, 2025
Scott Mutchnik :
2 p.m. in 427 SEO
Abstract
We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories.
Graduate students are particularly encouraged to attend.
We will finish showing that our example from last time, the generic projective planes of Conant and Kruckman, give nontrivial instances of finite F_Mb and the definable Morley property. We will then discuss connections between F_Mb and the definable Morley property, as well as connections between F_Mb and pseudolinearity.
Patrick Lutz :
3 p.m. in 636 SEO
Abstract
In his thesis, Arant introduced and studied the notion of Borel graphability: an equivalence relation E on a Polish space X is Borel graphable if there is a Borel graph on X whose connectivity relation is equal to E. In recent work, Tyler Arant, Alexander Kechris and I have addressed the question of which equivalence relations are Borel graphable. I will discuss some of our results, including on the Borel graphability of equivalence relations generated by Polish group actions, as well as mention some open questions.
Nov. 18, 2025
Scott Mutchnik :
2 p.m. in 427 SEO
Abstract
We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories.
Graduate students are particularly encouraged to attend.
We will introduce pseudolinearity and discuss the connections to F_Mb in the setting of simple theories.
John Baldwin :
3 p.m. in 636 SEO
Abstract
This paper provides two extensions of first order logic by ‘ω-rules’. In each
case we characterize the countable structures that have a categorical theory. In the one-
sorted inferential ω-logic, both Robinson’s system Q and Peano Arithmetic become cat-
egorical. In the two-sorted generalized ω-logic we show each Lω1,ω sentence defines the
same class of structures as a first-order theory with the appropriate ω-rule. These logics
are much weaker than any other proposed argument for the categoricity of arithmetic. The
results depend on proving that the inferential rules for the logics are categorical, i.e. they
uniquely determine certain truth-conditions for the logical connectives and quantifiers.
Nov. 25, 2025
Scott Mutchnik :
2 p.m. in 427 SEO
Abstract
We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories. Graduate students are particularly encouraged to attend.
Following our discussion of the relationship between F_Mb and the ranks of canonical bases, we will introduce k-linearity, a property defined by Buechler that quantifies the possible ranks of canonical bases of plane curves. After giving some examples, we will then state the pseudolinearity theorem of Buechler/Tomasic and Wagner, and potentially discuss the proof. We will conclude by talking about some problems for possible future discussion.
Sean Walsh :
3 p.m. in 636 SEO
Abstract
We characterize Martin-Löf randomness and Schnorr randomness in terms of the merging of opinions, along the lines of the Blackwell-Dubins Theorem. After setting up a general framework for defining notions of merging randomness, we focus on finite horizon events, that is, on weak merging in the sense of Kalai-Lehrer. In contrast to Blackwell-Dubins and Kalai-Lehrer, we consider not only the total variational distance but also the Hellinger distance and the Kullback-Leibler divergence. Our main result is a characterization of Martin-Löf randomness and Schnorr randomness in terms of weak merging and the summable Kullback-Leibler divergence. The main proof idea is that the Kullback-Leibler divergence between μ and ν, at a given stage of the learning process, is exactly the incremental growth, at that stage, of the predictable process of the Doob decomposition of the ν-submartingale L(σ)=−lnμ(σ)ν(σ). These characterizations of algorithmic randomness notions in terms of the Kullback-Leibler divergence can be viewed as global analogues of Vovk's theorem on what transpires locally with individual Martin-Löf μ- and ν-random points and the Hellinger distance between μ,ν. This is joint work with Simon Huttegger (UC Irvine) and Francesca Zaffora Blando (CMU). Preprint at: https://arxiv.org/abs/2504.00440
Dec. 2, 2025
Mariana Vicaria :
3 p.m. in 636 SEO
Abstract
In this talk, first I will present a very simplified proof of descent for stably dominated types in ACVF. I will also state a more general version of descent for stably dominated types in any theory, dropping the hypothesis of the existence of invariant extensions. This first part is work with Pierre Simon.
Later I will present a whole theory of residual domination for henselian valued fields of equicharacteristic zero. This is joint work with Pablo Cubides and Silvain Rideau Kikuchi. Among other things, we applied the original descent to prove a change of base statement for residual domination. We also show that in any henselian valued field (over an algebraically closed base), a global invariant type is residually dominated if and only if it is orthogonal to the value group, if and only if its reduct in ACVF is stably dominated. The results extend to valued fields with operators.
Scott Mutchnik :
2 p.m. in 427 SEO
Abstract
We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories.
Graduate students are particularly encouraged to attend.
We will discuss Dobrowolski, Kim and Ramsey's question of whether every NSOP_1 theory has the existence axiom, and the recent example, answering this question, of an NSOP_1 theory without the existence axiom. We will then discuss problems for further collaboration.
Dec. 9, 2025
Scott Mutchnik :
2 p.m. in 427 SEO
Abstract
We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories. Graduate students are particularly encouraged to attend.
We will discuss Dobrowolski, Kim and Ramsey's question of whether every NSOP_1 theory has the existence axiom, and the recent example, answering this question, of an NSOP_1 theory without the existence axiom. Continuing our discussion from last time about questions for possible future collaboration, we will conclude this seminar by giving some open problems on the existence axiom.
Jan. 20, 2026
Scott Mutchnik :
3 p.m. in 636 SEO
Abstract
Traditionally, we’ve thought of Shelah’s n-strict order property (or NSOP_n) hierarchy as being defined for positive integer values of n. However, this all changed recently due to the observation that the properties NSOP_r make sense even for non-integer values of r, including all real values of r not less than 3. It is open whether any of the properties NSOP_r for r real are distinct from all of Shelah’s original properties NSOP_n for integer values of n, raising the troubling, yet tantalizing, possibility that the classical classification-theoretic hierarchy is incomplete. Recent work demonstrates the complexity of this situation even at the purely combinatorial level.
In this talk, we discuss the implications of these new, real-valued strict order properties, and the questions surrounding them, for the more longstanding problems on the status of the classical classification-theoretic properties.
We first address the question of whether NSOP_2 is equal to NSOP_3. We discuss how the real-valued NSOP_r hierarchy reveals fine structure in between NSOP_2 and NSOP_3. Specifically, we show that it even makes sense to extend the definition of NSOP_r to r strictly in between 2 and 3, in the sense that NSOP_2 will still imply NSOP_r for r greater than 2. This observation applies an earlier result of the speaker than NSOP_1 is equal to NSOP_2.
We then turn to the question of whether every NTP_2, NSOP_(n + 1) theory is NSOP_n for n an integer at least 3. We give a precise sense in which, if the real-valued NSOP_r and integer-valued NSOP_n hierarchies coincide on sufficiently general grounds, this identity involving NTP_2 must be true. To a first approximation, this gives an apparent dichotomy between the possibility that the properties NSOP_r for real-valued r really are new properties of theories, and the resolution of main cases of the NTP_2-NSOP_n problem. If time permits, we will discuss even weaker, asymptotic conditions for the resolution of this problem, as well as the implications of the techniques involved for NTP_2 graph theory.
Jan. 27, 2026
Matthew Harrison-Trainor :
3 p.m. in 636 SEO
Abstract
Let T be a recursively axiomatizable first-order theory. We say that T is relatively decidable if, for any model of T, the atomic diagram of that model can compute the full elementary diagram. For example, if T is model complete, then there is a uniform decision procedure which works for any model of T. We characterize the complete relatively decidable theories by showing that they have a sort of conservative extension which is model complete. The proof combines a standard theorem from computable structure theory with an intricate but elementary model-theoretic argument.
Feb. 3, 2026
Scott Mutchnik :
2 p.m. in 427 SEO
Abstract
I will start to prove Saracino’s theorem, that every countably categorical theory has a model companion.
Feb. 10, 2026
Russell Miller :
3 p.m. in 636 SEO
Abstract
The absolute Galois group $\operatorname{Gal}(F)$
of a field $F$ is the Galois group of its algebraic closure $\overline{F}$
relative to $F$, containing precisely those automorphisms of $\overline{F}$
that fix $F$ itself pointwise. Even for a field as simple as the rational
numbers $\mathbb{Q}$, $\operatorname{Gal}(\mathbb Q)$ is a complicated
object. Indeed (perhaps counterintuitively), $\operatorname{Gal}(\mathbb Q)$
is among the thorniest of all absolute Galois groups normally studied.
When $F$ is countable, $\operatorname{Gal}(F)$ usually has the cardinality
of the continuum. However, it can be nicely presented as the set of all paths
through an $F$-computable finite-branching tree, built by a procedure
uniform in $F$. We will first consider the basic properties of this tree,
which depend in some part on $F$. Then we will address questions
about the subgroup consisting of the computable paths through
this tree, along with other subgroups
similarly defined by Turing ideals. One naturally asks to what
extent these are elementary subgroups of $\operatorname{Gal}(F)$
(or at least elementarily equivalent to $\operatorname{Gal}(F)$).
This question is connected to the computability of Skolem functions
for $\operatorname{Gal}(F)$, and also to the arithmetic complexity of
definable subsets of $\operatorname{Gal}(F)$. When $F=\mathbb Q$,
we have many questions and a few answers, partly due
to joint work with Debanjana Kundu.
In the simpler situations of the absolute Galois group of a finite field,
and of the Galois group of the cyclotomic field over $\mathbb Q$, much
more is known, thanks in part to joint work by Jason Block and the speaker.
Scott Mutchnik :
2 p.m. in 427 SEO
Abstract
We will aim to finish the proof of Saracino's theorem, that every countably categorical theory has a countably categorical model companion.
Feb. 17, 2026
James Freitag :
2 p.m. in 427 SEO
Abstract
Roughly speaking, a class of graphs is called nowhere dense if there is an upper bound on the size of a complete graph which occurs as an r-minor of an element of the class. This notion can also be characterized as there being a linear bound for the number of edges in terms of the number of vertices of any r-minor of an element of the class.
Nowhere density was introduced in the early 2000s, but was later noticed to be equivalent to the earlier notion of superflat, which is connected to model theory. In this talk, we will introduce the notions and explain the connections.
Feb. 24, 2026
Scott Mutchnik :
2 p.m. in 427 SEO
Abstract
Following our discussion of the proof of Saracino's theorem, stating that every countably categorical theory has a model companion, we give an application. We prove Bodor, Bodirsky and Marimon's recent theorem that, for any theory T, an existential formula exhibiting SOP_n in the model companion of T (when well-defined) exhibits SOP_n in T. We then use this to prove our result with Gabriel Day that, if NSOP_2 is equal to NSOP_3, then for any countably categorical theory T and formula φ(x,y) exhibiting SOP_2 in T, some ∃∀-formula in φ(x,y) exhibits SOP_3 in T.
Kyle Gannon :
3 p.m. in 636 SEO
Abstract
We investigate the semigroup of invariant types through the lens of Ellis theory; primarily focusing on definably amenable NIP groups. In the definably amenable NIP context, we observe that the collection of right strong $f$-generic types forms the unique minimal left ideal and thus, the Ellis subgroups are isomorphic to the $G/G^{00}$ via the canonical quotient map. As consequence of the Newelski-Pillay conjecture, the Ellis subgroups of the semigroup of invariant types are <i>abstractly isomorphic</i> to the Ellis subgroups of the semigroup of finitely satisfiable types in the definable amenable NIP setting. We are interested in the existence of <i>natural isomorphisms</i> from invariant Ellis subgroups to finitely satisfiable Ellis subgroups and we determine when these isomorphisms can be witnessed by (variants of) the canonical retraction map. We then provide several limiting examples. Outside of the NIP context, we provide an (dfg) abelian group in which the invariant Ellis subgroups and finitely satisfiable Ellis subgroups not isomorphic. This is joint work with Tomasz Rzepecki.
March 3, 2026
Yuyan He :
3 p.m. in 636 SEO
Abstract
A classical result of Ruzsa and Szemeredi shows that there is no subset A of the integers such that both the sumset A+A and the productset AA are small. Breuillard, Katz and Tao found a similar result for the subsets of finite prime fields of “medium size”. We show that under certain mild assumptions on a dimension theory , a “medium sized” subset of a field with small sumset and productset in the sense of not expanding in dimension indicates the existence of a subfield of the same dimension. This is joint work with Sergei Starchenko.
March 10, 2026
Yuki Takahashi :
3 p.m. in 636 SEO
Abstract
We discuss Chernikov’s conjecture that the burden is sub-additive. As partial progress toward this conjecture, we show that if T has a stronger version of dependent dividing (where dividing is witnessed by a formula in an existentially NIP reduct T_0 of T), then the burden agrees with the dp-rank witnessed by NIP formulas in T_0 and is thus sub-additive.
James Freitag :
2 p.m. in 427 SEO
Abstract
We will continue the results on nowhere dense graphs.
March 17, 2026
Zixuan Zhu :
3 p.m. in 636 SEO
Abstract
Let T_P be the theory of beautiful pairs of algebraically closed fields of fixed characteristic. It is known that for real tuples in models of T_P, SU-rank coincides with Morley rank and can be computed effectively. Building on Pillay's geometric description (2007) of imaginaries in T_P, we define an additive rank on all imaginaries, called the geometric rank. It takes values in ω*N+Z and coincides with SU-rank on real tuples. It refines SU-rank and characterizes forking in T_P^eq. As a consequence, we derive an explicit criterion for determining forking independence.
March 31, 2026
David Gonzalez :
3 p.m. in 636 SEO
Abstract
Scott rank measures the descriptive complexity of a countable structure. It has been precisely defined in many non-equivalent ways over the past several decades. Montalbán gave a definition of Scott rank about 10 years ago that has become standard in the literature because it is equivalent to many interesting measurements coming from various areas of logic. Later, he also defined the so-called parameterized Scott rank, which is his equivalent to unparameterized Scott rank after adding a parameter. This notion is equally robust. A lighthearted debate emerged about which rank is better, usually with the underlying assumption that it does not really matter which notion is used. This talk challenges this underlying assumption. In particular, we demonstrate the asymmetry of these notions in Ehrenfeucht theories and discuss how a counterexample to Vaught's conjecture would behave quite differently depending on which notion is preferred. We also discuss how to use our analysis to produce models where the unparameterized and parameterized Scott rank differ, addressing a question of Alvir, Csima, and MacLean. This is joint work with Dino Rossegger and Dan Turetsky.
April 21, 2026
Scott Mutchnik :
3 p.m. in 636 SEO
Abstract
Following the resolution of the longstanding formerly open question of whether the 1-strict order property, $\mathrm{SOP}_1$, is equal to $\mathrm{SOP}_2$, one of the main problems in model theory is to determine whether $\mathrm{SOP}_2$ is equal to $\mathrm{SOP}_3$. In this talk we discuss our very recent progress on this question, where we show, for $\mathcal{H}$ a hereditary class defined by finitely many forbidden weakly embedded substructures, that if the theory of every structure of which $\mathcal{H}$ is the age has $\mathrm{SOP}_2$, then the theory of every structure of which $\mathcal{H}$ is the age has $\mathrm{SOP}_3$. Crucially, this is a strict dichotomy: it is false if we replace $\mathrm{SOP}_2$ with the tree property (as demonstrated in examples of Conant and Kruckman, and Kruckman and Ramsey). We will start with some historical background on the problem of whether $\mathrm{NSOP}_2$ is equal to $\mathrm{NSOP}_3$, as well as the context for our theorem within the setting of $\mathrm{NSOP}_r$ theories for $r$ a real number. We will see how, by modifying recent work of Bodirsky, Bodor and Marimon, our general results reduce to what initially superficially appeared to be a mere verification for a concrete family of examples (the generic structures of Cherlin, Shelah and Shi). If time permits, we will give a very rough overview of the proof of this theorem, which involves harder versions of arguments on the real-valued $\mathrm{NSOP}_r$ hierarchy.
April 28, 2026
Jeremy Beard :
3 p.m. in 636 SEO
Abstract
Abstract elementary classes (AECs) provide an extension of first order model theory in which we can still attempt a classification theory. The question of when limit models (a kind of surrogate for saturated models for AECs) are isomorphic has connections to important open problems in AECs, such as Shelah's categoricity conjecture. Most work in this area is towards 'positive' results - that is, showing limit models are isomorphic. The question of when limit models are not isomorphic is less explored.
In this talk we give a full characterisation of the spectrum of limit models under reasonable assumptions in a stable AEC - that is, describe completely which limit models are isomorphic and which are not. In particular this applies to the first order stable setting. Given time we will discuss applications, a more general result in the 'positive' direction, and touch on a recent result which says that all high cofinality limit models are disjoint amalgamation bases. Based largely on joint work with Marcos Mazari-Armida.