Louise Hay Logic Seminar : Past Events
Past Seminars
The following seminars have already happened, you may instead view upcoming seminars in this series.
Oct. 9, 2008
Matthew Wright :
4 p.m. in SEO 712
Oct. 23, 2008
Fred Drueck :
4 p.m. in University of Chicago Ryerson 358
Nov. 6, 2008
Chris Conidis :
4 p.m. in SEO 1227
Nov. 20, 2008
James Freitag :
4 p.m. in University of Chicago RY 358
Jan. 22, 2009
Uri Andrews :
4:30 p.m. in University of Chicago TBA
Feb. 19, 2009
Kostya Slutsky :
4:30 p.m. in University of Chicago Ryerson 277
Abstract
Let G be a Polish group, that is a topological group which is also a completely metrizable separable topological space. We say that G has Rohlin property if there is an element g in G with dense conjugacy class. We'll discuss some examples of such groups and the connection between the Rohlin property and Model Theory of countable structures.
April 23, 2009
Fred Drueck :
4 p.m. in SEO 512
Abstract
A brief comparison of catagoricity transfer results in the rst or-
der case e.g Morley's Theorem and the Baldwin-Lachlin Theorem, versus the
abstract elementary case. In particular we will explore catagoricity trans-
fer results in the case of tame abstract elementary classes with arbitrarily
large models, amalgamation, joint embedding, and some tameness hypothe-
ses. In particular we will discuss results of Baldwin, Grossberg-VanDieren,
Grossberg- VanDieren-Villaveces, Lessman, and Shelah.
April 30, 2009
James Freitag :
4 p.m. in SEO 512
May 7, 2009
Sean Morris :
4 p.m. in SEO 512
Sept. 17, 2009
Elizabeth Gross :
3:30 p.m. in SEO 712
Oct. 22, 2009
James Freitag :
3:45 p.m. in SEO 712
Abstract
We will build (for arbitrary n) a strongly minimal structure in which the algebraic closure of any n elements is finite, but the algebraic closure of n+1 elements is infinite.
Geometry in the resulting structures will be discussed, should there be time.
Note the 3:45 start time, not 3:30.
Nov. 12, 2009
Uri Andrews :
3:30 p.m. in SEO 612
Jan. 21, 2010
James Freitag :
3 p.m. in SEO 612
Jan. 28, 2010
James Freitag :
3 p.m. in SEO 612
Abstract
We will continue to talk about ranks in differentially closed fields. I will also discuss a more geometric approach. This talk should make sense even without attending part 1.
Feb. 4, 2010
Phillip Wesolek :
3 p.m. in SEO 612
Abstract
We will discuss the model theory construction of the random graph and demonstrate its equivalence to the usual graph theory construction.
Feb. 11, 2010
Dave Sahota :
3 p.m. in SEO 612
Abstract
This talk will be an introduction to the basic techniques for studying O-minimal structures.
This will likely be useful material for those planning to attend the seminar next week when Michael Tychonievich (OSU) will talk.
Feb. 18, 2010
Michael Tychonievich :
3 p.m. in SEO 612
Abstract
Let a be a real number and let R be the expansion of the real field by the
logarithmic spiral {(t,t^{1+ia}): t > 0} and a. Then for any real b such
that a and b are linearly independent over the rationals, R defines no arc
of the power function. As a consequence, R defines no arc
of either the sine function or the exponential function.
Feb. 25, 2010
Charles Moss :
3 p.m. in SEO 612
March 4, 2010
Fred Drueck :
3 p.m. in SEO 612
Abstract
In an AEC there is, in general, no natural generalization of the notion of "forking" since there is not usually any syntactic structure to work with. However there is a notion of "splitting" which satisfies some of the proprieties of forking and, at least in some contexts, can be used to prove structure theorems about AECs.
One case in which splitting behaves very nicely is the case of atomic models (models realizing only isolated types) of a first order omega-stable theory.
April 1, 2010
Dave Sahota :
3 p.m. in SEO 612
April 8, 2010
Konstantin Slutsky :
3 p.m. in SEO 612
Abstract
The celebrated theorem of Gromov states that every group of
polynomial growth is virtually nilpotent. Gromov's original approach
used notion of a limit of metric spaces and had a strong flavor of
model theory. It was later polished and further developed by van den
Dries and Wilkie. They proved Gromov's theorem, and in fact a slightly
more general version of it, using machinery from non-standard
analysis. Both proofs had a "disadvantage": the heavy use of deep
results by Montgomery and Zippin on the Hilbert's fifth problem.
Recently a new proof, that avoids Montgomery-Zippin analysis, of
Gromov's result was suggested by Bruce Kleiner. I'll try to give an
overview of the ideas used in all the approaches above.
April 22, 2010
William Simmons :
3 p.m. in SEO 612
Abstract
We review some basic constructions in naive algebraic geometry that are
useful when applying model theory (e.g., the characteristic zero case of
the Mordell-Lang conjecture for function fields). The talk is
self-contained both with respect to algebra and logic.
April 29, 2010
Demirhan Tunc :
3 p.m. in SEO 612
Abstract
I will start with discussing decidability and undecidability results about expansions of Presburger Arithmetic, the first order theory of integers with addition. I will then describe how the real field with two small multiplicative groups gives rise to such an expansion and how we might approach this structure for obtaining a decidability result.
References:
[1] Bies A., A survey of arithmetical definability, A tribute to Maurice Boa, Bull. Belg. Math. Soc. Simon
Stevin 2001, suppl., 1-54.
[2] Point F., On decidable extensions of Presburger arithmetic: from A. Bertrand numeration systems to
Pisot numbers, Journal of Symbolic Logic, volume 65, number 3, 2000.
[3] Point, F., On the expansion (N;+; 2x) of Presburger arithmetic, To appear in Boolean Relation Theory
and Incompleteness, http://www.math.ohio-state.edu/friedman/manuscripts.html
[4] van den Dries, L., Gunaydin, A., The fields of real and complex numbers with a small multiplicative
group, Proc. London Math. Soc. (3), 93, 2006, no. 1, 43-81.
Sept. 9, 2010
Phillip Wesolek :
3 p.m. in SEO 612
Abstract
I prove a simplified version of Fraisse's Theorem by a topological argument. Viz. by first recognizing models as elements of Cantor space and then applying the Baire category theorem.
Sept. 16, 2010
Fred Drueck :
3 p.m. in SEO 612
Sept. 23, 2010
Lynn Scow :
3 p.m. in SEO 612
Abstract
We will discuss Shelah's proof that, in an NIP theory, a (type) definable group has a minimal
type-definable subgroup of bounded index. My explanation will derive from
the expositions in both Shelah's paper "Minimal bounded index subgroup for
dependent theories" and the exposition in "Groups, measures, and the NIP"
-- Hrushovski, Peterzil, Pillay.
Sept. 30, 2010
James Freitag :
3 p.m. in SEO 612
Abstract
We will discuss Buechler's paper "On nontrivial types of U-rank." Most of the talk will be devoted to proving Buechler's results in the simpler omega stable setting (where we can give simpler proofs). Before doing this, we will define forking, ranks, and geometries.
This is the first talk in a series regarding this paper and perhaps continuing with other stability theory.
A good backround for the talk would be understanding types, the Stone space and Morley rank.
Oct. 28, 2010
James Freitag :
3 p.m. in SEO 612
Abstract
We will continue to talk about nontrivial U-rank 1 types. The goal is to get through the first main result of the paper, having covered the back round material in the previous three meetings.
We will have tea to start.
Nov. 18, 2010
James Freitag :
3 p.m. in SEO 612
Abstract
We will continue discussing geometric stability theory. We will discuss Lascar's $\omega^\alpha$ theorem in particular after defining some notions from geometric stability theory. Then we will discuss specific instances of the theorem.
Jan. 20, 2011
Gabe Conant and Phil Wesolek :
3 p.m. in SEO 612
Feb. 3, 2011
Sam Ziegler :
3 p.m. in SEO 612
Feb. 10, 2011
Sam Ziegler and Gabe Conant :
3 p.m. in SEO 612
Feb. 17, 2011
Phillip Wesolek :
3 p.m. in SEO 612
Feb. 24, 2011
Phillip Wesolek :
3 p.m. in SEO 612
March 3, 2011
Phillip Wesolek :
3 p.m. in SEO 612
Abstract
We will begin discussion of "The Borel Complexity of Isomorphism for Theories with Many Types".
March 10, 2011
Gabe Conant and Fred Drueck :
3 p.m. in SEO 612
Abstract
Background will be given on Scott sentences and the equivalence relation $F_{2}$.
March 17, 2011
Gabe Conant :
3 p.m. in SEO 612
March 31, 2011
Sam Ziegler, Caroline Terry, and Phillip Wesolek :
3 p.m. in SEO 612
Abstract
This seminar will consist of three short talks covering the following background material: Coding countable structures as elements of a Polish space, the logic action, and the Feldman-Moore theorem and the Lopez-Escobar theorem.
April 11, 2011
Dave Sahota :
4 p.m. in SEO 712
April 14, 2011
Sam Ziegler and Phillip Wesolek :
3 p.m. in SEO 612
Abstract
The talks will cover Theorem 1.1 of the paper and the preliminary definitions and lemmas of section 2.
April 28, 2011
Phillip Wesolek :
3 p.m. in SEO 612
Abstract
We will cover section two of the paper.
Aug. 25, 2011
Phillip Wesolek :
3 p.m. in SEO 1227
Sept. 1, 2011
Phillip Wesolek :
3 p.m. in SEO 1227
Abstract
I will sketch the details of Willis' tidy subgroups and give an application to an old question of Halmos.
Sept. 15, 2011
Sam Alexander :
3 p.m. in SEO 1227
Abstract
We give an introduction to machine knowledge, and show that a truthful
machine which has basic reasoning and arithmetic abilities can
possibly know itself to be truthful, or know its own Godel number,
but not both. We then discuss well-foundedness of machine knowers:
Is there an infinite sequence of truthful machines,
each of which knows the Godel number and the truthfulness of the next? We show that this is not possible,
assuming the machines are capable of comprehending recursive ordinals.
Sept. 29, 2011
Tori Noquez :
3 p.m. in SEO 1227
Abstract
We will use a forcing argument to show that certain statements provable in a nonstandard extension of primitive recursive arithmetic are also provable in primitive recursive arithmetic.
Oct. 5, 2011
Caroline Terry :
3 p.m. in SEO 1227
Oct. 20, 2011
Gabe Conant and James Freitag :
3 p.m. in SEO 1227
Abstract
G. Conant will talk about why the Galois correspondence needs requires elimination of imaginaries (or a weak form).
J. Freitag will then discuss extending the classical Galois correspondence to extensions which are infinite degree (Poizat did this for the algebraic closure). After setting up the basics, we will introduce definable cocycles and continuous cocycles. The basics of Galois cohomology will be set up and some basic propositions will be proved.
Nov. 3, 2011
Phillip Wesolek :
3 p.m. in SEO 1227
Nov. 10, 2011
James Freitag :
3 p.m. in SEO 1227
Abstract
We will discuss Pillay's galois cohomology. The talk will aim towards a few basic results and will discuss potential generalizations; in particular, we will focus on the method of descent. We will assume only knowledge of galois theory.
The talk will be accessible to non model theorists.
Dec. 1, 2011
Gabe Conant and Phillip Wesolek :
3 p.m. in SEO 1227
Jan. 12, 2012
Gabe Conant and Phillip Wesolek :
3 p.m. in SEO 1227
Jan. 19, 2012
Gabe Conant :
3 p.m. in SEO 1227
Abstract
We continue discussing the model theory of the Rado and Henson graphs, and pursue a combinatorial understanding of forking and dividing in these theories.
Jan. 26, 2012
Phillip Wesolek :
3 p.m. in SEO 1227
Abstract
We discuss Vershik and Petrov's construction of an invariant measure via continuous graphs.
Feb. 2, 2012
Phillip Wesolek :
3 p.m. in SEO 1227
Feb. 9, 2012
Sam Ziegler :
3 p.m. in SEO 1227
Feb. 23, 2012
Tori Noquez :
3 p.m. in SEO 1227
Abstract
We will continue last week's discussion of Vaananen's paper Pseudo-finite model theory.
We will discuss the finite versions of results from model theory, such as compactness and Lowenheim-Skolem, which follow from the infinite versions. We will also see why the interpolation theorems do not hold in this framework, and a version of Lindstrom's Theorem for extensions of first order logic on pseudo-finite structures.
March 1, 2012
Lynn Scow :
3 p.m. in SEO 1227
Abstract
More on section 4 of the paper.
March 8, 2012
Caroline Terry :
3 p.m. in SEO 1227
March 13, 2012
Gabe Conant & Phil Wesolek :
4 p.m. in SEO 427
March 29, 2012
Phil Wesolek :
3 p.m. in SEO 1227
April 5, 2012
Gabe Conant :
3 p.m. in SEO 1227
April 12, 2012
Joe Zielinski :
3 p.m. in SEO 1227
April 19, 2012
Phil Wesolek & Sam Ziegler :
3 p.m. in SEO 1227
June 27, 2012
Andres Villaveces :
2 p.m. in SEO 1227
Abstract
Although the history of these dichotomies spans decades, recent
work by Baldwin, Larson, Hyttinen, Kolesnikov, Koerwien, Friedman, Grossberg, VanDieren and
myself(among others) has brought back some new ideas and techniques. I plan to
provide a description of one of the lines this research has taken - in particular, in connection
with various forms of the generalized Martin's Axiom.
June 29, 2012
Andres Villaveces :
11 a.m. in SEO 1227
Abstract
Theories without the independence property have been at the forefront of research in model theory (both in
generalizations of stability theory and in the breadth of applications they
have provided) during the past few years. Beyond First Order, very little
has been done. I will provide some new definitions, basic facts and questions. A bit will be added
on recent work on the Tree Property of the Second Kind.
Aug. 30, 2012
Gabe Conant and Phil Wesolek :
3 p.m. in SEO 427
Sept. 6, 2012
Phillip Wesolek :
3 p.m. in SEO 427
Abstract
Let $\Gamma$ be a vertex symmetric, locally finite graph and endow $Aut(\Gamma)$ with the topology of pointwise
convergence. It is easy to see $B(G):=\{g\in Aut(\Gamma):\exists n \forall v\in V\Gamma d(v,gv)\leq n\}$ is a subgroup.
Trofimov proved that $B(G)$ is also closed. In this talk
a new, elementary proof is given of Trofimov's result.
Sept. 13, 2012
Caroline Terry :
3 p.m. in SEO 427
Abstract
We introduce the notion of o-minimality, providing basic definitions and examples of o-minimal structures and theories. We sketch a proof of the Monotonicity theorem and state other basic results about o-minimal structures such as the uniform boundedness property and cell decomposition.
Sept. 20, 2012
Tori Noquez and Maxwell Levine :
3 p.m. in SEO 427
Abstract
We will discuss the algebra of ordered fields, and use this to show that the theory of real closed fields admits quantifier elimination, and thus, is model complete in the language of ordered rings.
From this, we prove that the theory of real closed fields is o-minimal, and give proofs of Hilbert's 17th problem and the Real Nullstellensatz.
Sept. 27, 2012
Gabe Conant :
3 p.m. in SEO 427
Oct. 4, 2012
Tori Noquez :
3 p.m. in SEO 427
Abstract
We will go over the proof of the main theorem in Pila and Wilkie's 2006 paper The Rational Points of a Definable Set.
It says that for $X$, a subset of $\mathbb R^n$, which is definable in an o-minimal expansion of $\mathbb R$, there are (in a suitable sense) very few rational points of $X$ which do not lie on some connected semialgebraic subset of $X$ of positive dimension.
Oct. 11, 2012
Ahuva Shkop :
3 p.m. in SEO 427
Oct. 18, 2012
Sam Ziegler :
3 p.m. in SEO 427
Abstract
In this talk, we prove that the logic action of $S_{\infty}$ on the space of countable
graphs is universal, in the sense that any $S_{\infty}$ orbit equivalence relation Borel
reduces to the orbit equivalence relation on graphs.
Oct. 25, 2012
Joe Zielinski :
3 p.m. in SEO 427
Nov. 1, 2012
Scott Cramer :
3 p.m. in SEO 427
Abstract
I will give an introduction to the large cardinal hierarchy and the axioms at the very top of this hierarchy. In particular I will demonstrate how these axioms might relate to notions in descriptive set theory. I'll also consider whether any or all of these axioms might or might not be inconsistent
Nov. 8, 2012
Phillip Wesolek :
3 p.m. in SEO 427
Abstract
We give an account of the following theorem of Jarden and Lubotzky: Let $G$ and $H$ be finitely generated profinite groups.
If $G\equiv H$ in the language of groups, then $G\cong H$.
Nov. 15, 2012
Gabe Conant :
3 p.m. in SEO 427
Abstract
Given $T$, an NIP theory, and $G$, a group definable in a sufficiently saturated model of $T$, we prove the existence of a minimal type-definable subgroup of $G$ of bounded index.
Nov. 29, 2012
Caroline Terry :
3 p.m. in SEO 427
Abstract
It is a theorem due to Shelah that for L a countable language, T a stable theory, then if T has a prime model over a set A, it is unique. In this talk we describe two examples where this is not the case. In particular we describe an example of a stable theory in an uncountable language with non-isomorphic prime models, and an example of an unstable theory in a countable language with non-isomporphic prime models over a set A.
Jan. 17, 2013
Gabe Conant & Phil Wesolek :
3 p.m. in SEO 427
Jan. 24, 2013
Tori Noquez :
3 p.m. in SEO 427
Abstract
In preparation for our discussion of strong minimality in continuous logic, we will review some of the basics of continuous logic. In particular, we will review definability of sets and the properties of algebraic closure.
Jan. 31, 2013
Tori Noquez :
3 p.m. in SEO 427
Abstract
We will continue last week's discussion and talk about why there does not appear to be a good notion of strong minimality for continuous logic.
We will look at Isaac Goldbring's example of a notion of strong minimality for continuous logic for which some facts from classical logic hold. We'll see though that it does not appear to give many interesting examples of strongly minimal structures.
Feb. 7, 2013
Phillip Wesolek :
3 p.m. in SEO 427
Abstract
We discuss the basic theory of the Urysohn space and give an account of Goldbring's <i>Definable functions in Urysohn's metric space </i>.
Feb. 14, 2013
Caroline Terry :
3 p.m. in SEO 427
Abstract
We define model theoretic algebraic independence, providing examples within classical model theory and proving it satisfies full existence in this setting. We then discuss why this proof cannot be immediately adapted to the setting of continuous model theory.
Feb. 21, 2013
Gabe Conant :
3 p.m. in SEO 427
Feb. 28, 2013
Jin Du :
3 p.m. in SEO 427
Abstract
Let [tex]X[/tex] be a Polish space and [tex]\{f_{\alpha}\}_{\alpha}[/tex] an increasing sequence of functions from [tex]X[/tex] into the real numbers. I will discuss how imposing measurability conditions on the [tex]f_{\alpha}[/tex]'s affects whether or not the sequence can be uncountable.
March 7, 2013
Sam Ziegler :
3 p.m. in SEO 427
Abstract
We survey a paper of Ben Miller in which the author proves various dichotomy theorems
of descriptive set theory using graphs. These proofs are "classical," as opposed to Harrington's
proof of Silver's theorem, which uses effective methods (for example). If there is time,
other notions from Borel graph theory will be discussed.
March 14, 2013
Aida Alibek :
3 p.m. in SEO 427
Abstract
The number of countable non-isomorphic models of a weakly
o-minimal theory is considered. Effective construction of examples of a weakly
o-minimal theory with a finite and a countable number of countable models, with
application of notions of type orthogonality and quasi-follower.
March 21, 2013
Tori Noquez :
3:30 p.m. in SEO 427
Abstract
We will discuss the result of Janak Ramakrishnan's paper "Definable Linear Orders Definably Embed Into Lexicographic Orders in o-minimal Structures". If (P,<) is M-definable in an o-minimal group M with n=dim(P), then there is an M-definable embedding of (P,<) into (M^{2n+1},<_lex). We will go over the proof of the n=1 case during this talk.
April 11, 2013
Maxwell Levine :
3 p.m. in SEO 427
Abstract
We discuss Tennenbaum's Theorem, which states that any countable nonstandard model of Peano arithmetic is nonrecursive, as well as its connection to the Godel-Rosser Theorem and its implications for the study of Peano arithmetic.
April 23, 2013
Maxwell Levine :
4 p.m. in SEO 427
Abstract
We discuss Tennenbaum's Theorem, which states that any countable nonstandard model of Peano arithmetic is nonrecursive, as well as its connection to the Godel-Rosser Theorem and its implications for the study of Peano arithmetic.
May 2, 2013
Gabe Conant :
3 p.m. in SEO 427
Abstract
We will define and discuss five notions of independence, which are frequently studied in an arbitrary first order theory. After assuming the theory in question has the property that algebraic closure satisfies exchange, we will add a sixth notion of dimensional independence, and see how it fits in with the others. Finally, we prove the equivalence of all six notions in strongly minimal theories with infinite algebraic closure. Altogether, this gives a characterization of independence that avoids any explicit mention of Morley rank or stability, but still demonstrates much of the same good behavior.
Aug. 29, 2013
Gabe Conant and Phillip Wesolek :
4 p.m. in SEO 427
Sept. 5, 2013
Maxwell Levine :
4 p.m. in SEO 427
Abstract
This talk is intended to introduce first-year graduate students to concepts vital to current research in set theory, placing a particular emphasis on independence results. In addition to more elementary concepts not often taught to undergraduates (cofinality, club sets, stationary sets, the arithmetical hierarchy, etc.) we will discuss the construction of inner models and forcing extensions as well as strengthenings of ZFC.
Sept. 12, 2013
Joseph Zielinski :
4 p.m. in SEO 427
Abstract
The second installment in our series of introductory talks, we outline the history, techniques, results, trends, and applications of descriptive set theory.
Sept. 19, 2013
Sam Ziegler & Phil Wesolek :
4 p.m. in SEO 427
Abstract
Sam will speak about the relation of weak equivalence between actions of discrete groups on the interval [0,1],
in particular the non-classifiability of this relation.
Phil will speak about the following: many closed subgroups of S_{\infty} admit a comeagre conjugacy class. For example, S_{\infty} (folklore)
or Aut(\mathbb{Q},<) (Truss) admit a comeagre conjugacy class. Kechris and Rosendal ask if a locally compact
subgroup of S_{\infty} can admit a comeagre conjugacy class. We answer the question in the negative via an
analysis of locally compact subgroups of S_{\infty} with topological conditions on a conjugacy class.
Sept. 26, 2013
Gabriel Conant :
4 p.m. in SEO 427
Oct. 3, 2013
Caroline Terry :
4 p.m. in SEO 427
Oct. 10, 2013
Victoria Noquez :
4 p.m. in SEO 427
Abstract
Abstract: We will be discussing Dolich, Goodrick, and Lippel's paper Dp-Minimality: Basic Facts and Examples. In particular, we'll focus on the basic facts and relationships with similar notions, such as VC-minimality and weak o-minimality.
Oct. 17, 2013
Gabriel Conant :
4 p.m. in SEO 427
Oct. 24, 2013
Will Boney :
4 p.m. in SEO 427
Abstract
We present a nonforking relation for type short and tame Abstract Elementary Classes that generalizes the first order notion of coheir (finite satisfiability) in stable, first order theories. We identify sufficient conditions for this nonforking to be well-behaved (with a set-theoretic aside) and give applications to the uniqueness of limit models and superstability.
Oct. 31, 2013
Jin Du :
4 p.m. in SEO 427
Nov. 7, 2013
Martin Zeman :
4 p.m. in SEO 427
Abstract
Absoluteness of statements between transitive models of set theory is a well-studied topic. The classical result in this area is the well-known Shoenfield absoluteness theorem stating that every \Sigma^1_2 statement is absolute in this sense. Absoluteness for statements of higher complexity requires large cardinals. By classical work of Martin and Solovay, the existence of proper class of measurable cardinals guarantee that \Sigma^1_3 statements are absolute for generic extensions. The well-known high-level result in the area is Woodin's theorem on the generic absoluteness of \Sigma^2_2 statements. There are other types of absoluteness results which concern models that are not generic extensions. Among the most well-known are results on \Sigma^1_3 correctness of the core model, where the classical version is due to Jensen and the modern version is due to Steel. In the talk we focus on the absoluteness of \Sigma^1_3 statements with respect to any extension, not just generic, which correctly computes the successor of at least one regular cardinal. Under the suitable large cardinal hypothesis we show that such an extension is \Sigma^1_3 absolute. This is a joint work with Andres Caicedo.
Nov. 14, 2013
Maxwell Levine :
4 p.m. in SEO 427
Nov. 21, 2013
Austin Yim :
4 p.m. in SEO 427
Abstract
In model theory, Galois correspondences arise in theories and their models which have a property similar to the elimination of imaginaries. The classical understanding of algebraic Galois theory can be recovered in the model-theoretic context using the appropriate and relevant theories of fields, but all theories of fields carry this requisite property. A Galois theory of exponential fields therefore exists, though not comprehensible with the model theory of complex exponentiation. The theory of psuedoexponentiation addresses some, though not all, of these difficulties, enough to characterize the algebraic subfield of the prime substructure. Using the Kronecker-Weber theorem as inspiration, some results about the abelian extensions of the prime substructure are obtained, primarily by developing a field-theoretic understanding of the algebraic subfield.
Dec. 5, 2013
Sam Ziegler :
4 p.m. in SEO 427
Abstract
Any Polish group can be equipped with a compatible left-invariant metric; however, there may be no compatible
left-invariant metric which is also complete. Groups which do have such a metric are known as CLI groups. In this talk, we will
examine a theorem, due to Gao, characterizing when groups of the form $Aut(\mathcal{M})$ are CLI. It turns out that for such automorphism groups, CLI can be
reformulated as a property of the Scott sentence of $\mathcal{M}$.
Jan. 16, 2014
Sam Ziegler & Joseph Zielinski :
4 p.m. in SEO 427
Jan. 23, 2014
Gabriel Conant :
3 p.m. in SEO 427
Abstract
We define dividing lines in first order theories that create a region ripe with open questions and interesting examples, in particular: infinitely parameterized equivalence relations, $\omega$-free pseudo-algebraically closed fields, and infinite dimensional vector spaces with a bilinear form.
Jan. 30, 2014
Joseph Zielinski :
3 p.m. in SEO 427
Feb. 6, 2014
Phillip Wesolek :
3 p.m. in SEO 427
Abstract
The class of constructible t.d.l.c. Polish groups is the class of t.d.l.c. Polish groups which are built from profinite and discrete
groups by group extension and countable increasing union. We first explore closure properties of this class. We then use these
closure properties to show the following fact: if there is a surjectively universal t.d.l.c. Polish group, then there is a constructible
group which is surjectively universal for the class of constructible groups. This fact is a step in the direction of answering the following: (Gao) Is there a surjectively universal t.d.l.c. Polish group?
Feb. 13, 2014
Maxwell Levine :
3 p.m. in SEO 427
Feb. 27, 2014
Amit Shah :
3 p.m. in SEO 427
March 6, 2014
Jonathan Wolf :
3 p.m. in SEO 427
March 13, 2014
Bruno de Mendonca Braga :
3 p.m. in SEO 427
March 20, 2014
Michael Lieberman :
3 p.m. in SEO 427
Abstract
We discuss recent joint work with Jiri Rosicky which seeks to extend a fragment of the classification theory of AECs to the more general framework of accessible categories, particularly for accessible categories with directed colimits: essentially AECs minus coherence. There are several pleasant surprises---a generalization of Boney's recent theorem on tameness under a large cardinal hypothesis follows from work of Makkai and Pare, and these categories admit a robust Ehrenfeucht-Mostowski functor which can be used to mimic certain constructions in AECs. On the other hand, this investigation also illuminates the areas and arguments where coherence is indispensable...
April 3, 2014
Jin Du :
3 p.m. in SEO 427
April 10, 2014
Sam Ziegler :
3 p.m. in SEO 427
Abstract
Topological similarity, introduced by Rosendal, is an invariant of conjugacy in Polish groups. In this talk, we will define this invariant
and discuss its applications.
May 1, 2014
Gabriel Conant :
3 p.m. in SEO 427
Abstract
We construct the universal Polish ultrametric space with rational spectrum, and then prove some model theoretic properties of its theory.
Aug. 28, 2014
Caroline Terry :
4 p.m. in SEO 427
Sept. 4, 2014
Caroline Terry :
4 p.m. in SEO 427
Abstract
We introduce the notion of a logical 0-1 law for classes of finite first order structures. We then sketch the proof that the class of finite triangle-free graphs has a logical 0-1 law.
Sept. 11, 2014
Aida Alibek :
4 p.m. in SEO 427
Abstract
We will consider the example of a complete countable theory (due to B. Omarov), whose expansions reduce the number of countable non-isomorphic models. We will see the reduction from $\omega$ to $n$, as well as $2^\omega$ to $\omega$ and $2^\omega$ to $n$.
Sept. 18, 2014
Maxwell Levine :
4 p.m. in SEO 427
Abstract
We will discuss Laver's result that the ground model is always definable with one parameter from the point of view of any forcing extension. This result, despite seeming very essential, came about long after forcing was discovered. This talk will go through the proof in a gently technical manner.
Sept. 25, 2014
Jin Du :
4 p.m. in SEO 427
Oct. 2, 2014
Hunter Chase :
4 p.m. in SEO 427
Oct. 9, 2014
Joseph Zielinski :
4 p.m. in SEO 427
Oct. 23, 2014
Sam Ziegler :
4 p.m. in SEO 427
Abstract
Scott sets are exactly the standard systems of nonstandard models of PA.
Oct. 30, 2014
Tori Noquez :
4 p.m. in SEO 427
Nov. 6, 2014
Jonathan Wolf :
4 p.m. in SEO 427
Nov. 20, 2014
Jon Yaggie :
4 p.m. in SEO 427
Abstract
I will give a brief introduction to finite model theory and belief revision.
This talk will provide the background for the next talk (to be given
in two weeks) on applying finite model theory to prove
results on characterizability in belief revision.
Dec. 4, 2014
Jon Yaggie :
4 p.m. in SEO 427
Abstract
I will give a brief introduction to finite model theory and belief revision. This talk will be on applying finite model theory to prove results on characterizability in belief revision.
Jan. 15, 2015
Gianluca Paolini :
4 p.m. in SEO 427
Abstract
We prove that the form of conditional independence at play in database theory and independence logic is reducible to the first-order dividing calculus in the theory of atomless boolean algebras. This establishes strong connections between independence in database theory and stochastic independence. As indeed, in light of the aforementioned reduction and recent work of Ben-Yaacov, the latter case of independence can be seen as the measure-theoretic version of the former.
Jan. 22, 2015
Caroline Terry :
4 p.m. in SEO 427
Jan. 29, 2015
Maxwell Levine :
4 p.m. in SEO 427
Feb. 5, 2015
Hunter Chase :
4 p.m. in SEO 427
Abstract
We will discuss either Emily Post's theorem on the relationship between epistolography and dining etiquette, or Emil Post's theorem on the relationship between arithmetical hierarchy and Turing degrees.
Feb. 12, 2015
Sam Ziegler :
4 p.m. in SEO 427
Feb. 19, 2015
Joseph Zielinski :
4 p.m. in SEO 427
Feb. 26, 2015
Victoria Noquez :
4 p.m. in SEO 427
Abstract
Keisler first introduced the notion of the randomization of a structure M: a new structure whose universe is a set of "random elements" of M. It was later discovered that it is natural to consider Keisler's randomizations in the framework of continuous logic, and in this way, a class of continuous structures arises naturally from existing first order structures. As such, for a classical first order theory T, we get a continuous theory T^R, whose models are essentially spaces of M-valued random variables, where M is a model of T.
In this talk we will discuss the technical considerations required to consider randomizations as continuous structures and some of the properties of theories which are preserved. If time permits, we will discuss Ben Yaacov's proof that randomizing a theory preserves (the continuous analog of) NIP.
March 5, 2015
Caroline Terry :
4 p.m. in SEO 427
March 12, 2015
Tatyana Zambarnaya :
4 p.m. in SEO 427
Abstract
We consider a condition, which is similar to a binary relation of following.
We prove that a small theory, which has a quasi-successor formula, has the maximum number of countable non-isomorphic models.
Nargiz Tazabekova :
4:30 p.m. in SEO 427
Abstract
We consider quasi-neighborhoods, neighborhoods, their properties and connection with each other. And then show how neighborhoods can affect the number of countable models. Also we are going
to describe how orthogonality of types is connected with the definability of quasi-neighborhoods.
March 19, 2015
Olzhas Umbetbayev :
4 p.m. in SEO 427
Abstract
TBA
Aisha Yershigeshova :
4:30 p.m. in SEO 427
April 2, 2015
Aida Alibek :
4 p.m. in SEO 427
April 9, 2015
Gabriel Conant :
4 p.m. in SEO 427
Abstract
We consider complete first-order theories equipped with an axiomatic ternary relation designed to capture the behavior of free amalgamation in relational languages. We will discuss model theoretic consequences of the existence of such a ternary relation, and also give several examples, each of which is a well-known generic structure produced by a Fraïssé or Hrushovski construction.
April 30, 2015
Jonathan Wolf :
4 p.m. in SEO 427
Aug. 27, 2015
Caroline Terry and Joseph Zielinski :
4 p.m. in SEO 427
Sept. 3, 2015
Maxwell Levine :
4 p.m. in SEO 427
Sept. 10, 2015
Victoria Noquez :
4 p.m. in SEO 427
Sept. 17, 2015
Noah Schoem :
4 p.m. in SEO 427
Abstract
This talk will give an overview of Reverse Mathematics.
We will discuss the five main systems
$RCA_0$, $WKL_0$, $ACA_0$, $ATR_0$, and $\Pi^1_1-CA_0$ of Reverse Mathematics and
present some theorems that are equivalent to these systems.
Sept. 24, 2015
Joseph Zielinski :
4 p.m. in SEO 427
Oct. 1, 2015
McKinley Meyer :
4 p.m. in SEO 427
Oct. 15, 2015
Zachary Fox :
4 p.m. in SEO 427
Oct. 22, 2015
Hunter Chase :
4 p.m. in SEO 427
Oct. 29, 2015
Aida Alibek :
4 p.m. in SEO 427
Abstract
In 1961 R. Vaught proposed a conjecture that the number of countable models of a first-order complete theory in a countable language is finite, $\aleph_0$ or $2^{\aleph_0}$.
The talk will be an overview of various results, connected to this open problem. We will consider an especially nice theorem of L. Mayer for o-minimal theories. We will also talk about some results for the topological Vaught conjecture.
Nov. 5, 2015
Jin Du :
4 p.m. in SEO 427
Nov. 19, 2015
Shehzad Ahmed :
4 p.m. in SEO 427
Abstract
We say that a cardinal $\lambda$ is a Jónsson cardinal if it satisfies the following weak Ramsey-type property: given any coloring
$F:[\lambda]^{<\omega} \rightarrow \lambda$ of the finite subsets of $\lambda$ in $\lambda$-many colors, there exists a set
$H \in [\lambda]^\lambda$ such that the range of $F \restriction [H]^{<\omega}$ is a proper subset of $\lambda$. One of the
big driving forces present in early chapters Cardinal Arithmetic is an attempt to understand the combinatorial structure at
and around Jónsson cardinals. The goal of this talk is to highlight the connection between Jónsson cardinals and various forms
of club guessing, which can be thought of as weakenings of $\diamond$ principles. Our focus will be on what happens at successors
of singulars, and I will attempt to explain why the following question is difficult: "Is it consistent that there exists a singular
cardinal $\mu$ such that $\mu^+$ is Jónsson?"
Dec. 3, 2015
Jonathan Wolf :
4 p.m. in SEO 427
Jan. 14, 2016
Aida Alibek, Maxwell Levine :
4 p.m. in SEO 427
Jan. 21, 2016
Hunter Chase :
4 p.m. in SEO 427
Jan. 28, 2016
McKinley Meyer :
4 p.m. in SEO 427
Feb. 4, 2016
Noah Schoem :
4 p.m. in SEO 427
Abstract
TBD
Feb. 11, 2016
Jonathan Wolf :
4 p.m. in SEO 427
Feb. 18, 2016
Aida Alibek :
4 p.m. in SEO 427
Abstract
In 1944 Emil Post posed a problem, whether there is any r.e. degree strictly between 0 and 0′. This problem was solved independently by Friedberg and Muchnik in the 1950's.
We will learn the notion and some results about simple sets. They will be crucial in a few weeks, when we will look at the proof of Post's problem.
March 3, 2016
Jin Du :
4 p.m. in SEO 427
March 10, 2016
Maxwell Levine :
4 p.m. in SEO 427
March 31, 2016
Victoria Noquez :
4 p.m. in SEO 427
April 28, 2016
Roland Walker :
4 p.m. in SEO 427
Abstract
Recursive saturation and $S$-saturation are interesting concepts at the intersection of model theory and recursion theory.
In this talk, we will define and discuss recursively saturated models, Scott sets, effectively perfect theories, standard systems, and $S$-saturated models.
We will prove that any recursively saturated model of an effectively perfect theory is $S$-saturated for exactly one Scott set $S$.
It is natural to ask if all Scott sets arise in this fashion.
In 1982, Knight and Nadel showed that all Scott sets of size at most $\aleph _1$ arise as the standard system of some nonstandard model of Peano Arithmetic.
We will prove a more general version of this result which applies to any computable theory.
This closes the question under CH.
We will discuss recent progress which closes the question for specific theories when CH is not assumed.
Aug. 25, 2016
Maxwell Levine and Aida Alibek :
4 p.m. in SEO 427
Sept. 8, 2016
Noah Schoem :
4 p.m. in SEO 427
Abstract
After a quick review of the apparatus of forcing, we will cover how the generic extension
$M[G]$ is built from interpreted $\mathbb{P}$-names.
We will then cover cardinal collapse with $Col(\omega,\kappa)$
and the consistency of $2^{\aleph_0}=\aleph_2$ from $Add(\omega,\omega_2)$.
Sept. 15, 2016
Aida Alibek :
4 p.m. in SEO 427
Abstract
This is going to be an introductory talk to next weeks seminar on Externally Definable Sets and Shelah Expansions.
We will go over such important notions of model theory as types, monster model, heirs and coheirs, indiscernible sequence and NIP theories.
Sept. 22, 2016
Roland Walker :
4 p.m. in SEO 427
Abstract
A major focus of model theory is the study of definable sets. In this talk we will discuss externally definable sets and Shelah expansions. Our goal will be to prove a theorem of Saharon Shelah stating that in the NIP setting, Shelah expansions have quantifier elimination. While proving the theorem, we will review and use several basic techniques involving quantifier-free types, heirs, coheirs, and coheir sequences. We will also discuss some areas of open research concerning externally definable sets.
Oct. 6, 2016
Carl Dean :
4 p.m. in SEO 427
Abstract
TBA
Oct. 13, 2016
Jonathan Wolf :
4 p.m. in SEO 427
Oct. 27, 2016
Hunter Chase :
4 p.m. in SEO 427
Nov. 3, 2016
Victoria Noquez :
4 p.m. in SEO 427
Nov. 10, 2016
Jin Du :
4 p.m. in SEO 427
Nov. 17, 2016
Noah Schoem :
4 p.m. in SEO 427
Abstract
Cohen (1963) proved $Con(ZFC)\implies Con\left(ZFC+2^{\aleph_0}=\aleph_2\right)$ using the method of forcing.
Further work showed that $Con(ZFC)\implies Con\left(ZFC+2^{\aleph_0}=\kappa\right)$ for any $\kappa$ of uncountable cofinality.
This raises the question of how the continuum function $\kappa\mapsto 2^\kappa$ can behave,
which has been answered for regular cardinals (Easton 1970).
We will cover Easton's Theorem and the techniques required to prove it, namely class-sized forcing
and product forcing with Easton support.
Dec. 1, 2016
Hunter Chase :
4 p.m. in SEO 427
Abstract
We review the fundamentals of $\omega$-stable groups and show that every infinite $\omega$-stable field is algebraically closed.
Jan. 12, 2017
Aida Alibek and Maxwell Levine :
4 p.m. in SEO 427
Jan. 26, 2017
Maxwell Levine :
4 p.m. in SEO 427
Abstract
The development of mathematical logic in the twentieth century was characterized by a number of negative results: The Incompleteness Theorems, the MDRP theorem, the nonexistence of solutions to the Halting Problem, and so on. Arrow's Theorem, which essentially asserts that there is no fair voting system, appears to fall into this mold. In this talk I will present Arrow's Theorem and discuss its relationship to mathematical logic.
Feb. 2, 2017
Aida Alibek :
4 p.m. in SEO 427
Feb. 16, 2017
Jin Du :
4 p.m. in SEO 427
Feb. 23, 2017
Matthew DeVilbiss :
4 p.m. in SEO 427
Abstract
TBA
March 2, 2017
Noah Schoem :
4 p.m. in SEO 427
March 9, 2017
Hunter Chase :
4 p.m. in SEO 427
Abstract
We will discuss forking and other notions of independence. We will discuss combinatorial notions that are useful in classifying theories. We will discuss how forking and other notions of independence behave in these theories, with the aim of showing that forking equals dividing over nicely behaved sets (including models) in NTP2 theories.
April 6, 2017
Jonathan Wolf :
4 p.m. in SEO 427
April 20, 2017
Jin Du :
4 p.m. in SEO 427
April 27, 2017
Noah Schoem :
4 p.m. in SEO 427
Abstract
Although mostly an idle curiosity,
the quest to name large natural numbers gives rise to, and requires substantial use of,
tools from mathematical logic.
We begin with elementary attempts and conclude with state-of-the-art techniques
that go beyond $ZFC$.
Aug. 31, 2017
Noah Schoem, Hunter Chase, Jin Du :
4 p.m. in SEO 427
Sept. 7, 2017
Hunter Chase :
1 p.m. in SEO 427
Abstract
Most model-theoretic techniques are geared toward infinite structures. We explore some of the challenges, techniques, and questions that arise when dealing with finite structures instead.
No background in model theory is assumed.
Sept. 14, 2017
Thomas Dean :
1 p.m. in SEO 427
Sept. 21, 2017
Roland Walker :
1 p.m. in SEO 427
Abstract
Vapnik-Chervonenkis dimension and density are two measures of combinatorial complexity which arose from the study of probability theory. During this two-part talk, we will discuss these measures and their duals both in the classical and model-theoretic contexts, prove the famous Sauer-Shelah Lemma, discuss the relationship between VC dimension and NIP, and time permitting discuss some recent applications and open questions.
Sept. 28, 2017
Roland Walker :
1 p.m. in SEO 427
Abstract
Vapnik-Chervonenkis dimension and density are two measures of combinatorial complexity which arose from the study of probability theory. During this two-part talk, we will discuss these measures and their duals both in the classical and model-theoretic contexts, prove the famous Sauer-Shelah Lemma, discuss the relationship between VC dimension and NIP, and time permitting discuss some recent applications and open questions.
Oct. 5, 2017
Noah Schoem :
1 p.m. in SEO 427
Abstract
$0^#$ is a fragment of large cardinals whose non-existence says that the constructible universe $L$ is very close to $V$.
Jensen's Covering Lemma makes this precise; we will explore some of its consequences and sketch a proof.
Oct. 12, 2017
Hunter Chase :
1 p.m. in SEO 427
Abstract
We review the ultrapower construction and describe the tendencies of ultrapowers to saturate models. We then define Keisler's order and discuss its structure.
Oct. 26, 2017
Matthew DeVilbiss :
1 p.m. in SEO 427
Nov. 2, 2017
Noah Schoem :
4 p.m. in SEO 427
Nov. 9, 2017
Thomas Dean :
1 p.m. in SEO 427
Nov. 16, 2017
Jonathan Wolf :
1 p.m. in SEO 427
Nov. 30, 2017
Aida Alibek :
1 p.m. in SEO 427
Abstract
In 1993 the ASL Committee on Logic Education has produced an official Guideline for Logic Education. In this seminar we will discuss what the guidelines say about undergraduate education and how we, as future educators can use them to improve our teaching of logic.
This will be a seminar/discussion, where time permitting, we will consider some examples of logic education at various institutions.
Dec. 7, 2017
Jin Du :
1 p.m. in SEO 427
Jan. 18, 2018
Noah Schoem, Hunter Chase :
4 p.m. in SEO 427
Jan. 25, 2018
Hunter Chase :
4 p.m. in SEO 427
Abstract
We explain some fundamentals of machine learning and some connections with model theory. Joint with J. Freitag.
Feb. 8, 2018
Alexander Berenbeim :
4 p.m. in SEO 427
Abstract
The surreal numbers are a real closed field that are defined as a binary tree whose depth is the class of ordinals. This introduction to the surreal numbers goes over the inductive construction of the surreal numbers, as well as the definition of the arithmetic, exponential and logarithmic operations.
Feb. 15, 2018
Thomas Dean :
4 p.m. in SEO 427
Abstract
We discuss how appropriately defined elementary submodels of $H_\theta$ can be used to prove results in general topology. We prove for example a famous result by Arhangel'skii that first countable compact Hausdorff spaces have cardinality at most continuum.
Feb. 22, 2018
Thomas Dean :
4 p.m. in SEO 427
March 1, 2018
Noah Schoem :
4 p.m. in SEO 427
March 8, 2018
Noah Schoem :
4 p.m. in SEO 427
Abstract
Continuing on last week's theme, where we presented Solovay's original proof
of the Stationary Splitting Theorem,
we will continue examining the method of ultrapowers generated by
filters that are $V$-ultra over a ground model $V$.
We focus on a $V$-generic filter generated by a Lévy collapse to prove
Silver's Theorem that the first failure of the SCH cannot occur at a singular cardinal of uncountable cofinality.
March 15, 2018
Matthew DeVilbiss :
4 p.m. in SEO 427
Abstract
This talk will be a basic introduction to the correspondence between topological dynamics and model theory in the context of definable groups.
March 22, 2018
Matthew DeVilbiss :
4 p.m. in SEO 427
Abstract
In the infancy of model theory, Tarski proved that elementary Euclidean geometry is decidable, i.e. there is an algorithm that decides the truth or falsehood of all first-order sentences pertaining to this geometry. While this is fairly well known, it seems the proof is far more obscure. In this talk, I will elaborate on this proof.
Depending on how that goes, I might also talk about elementary hyperbolic.
April 5, 2018
Mihail Hurmuzov :
4 p.m. in SEO 427
April 12, 2018
Hunter Chase :
4 p.m. in SEO 427
Abstract
VC-dimension and Shelah's 2-rank (also known as Littlestone or thicket dimension) are important measures of complexity associated with the NIP/IP and stable/unstable dividing lines. We examine these through the lens of binary trees and sequences. We introduce the concept of banned sequence problems as a tool to obtain new proofs of the Sauer-Shelah lemma and some of its variants. We use these results to modestly improve a result of Malliaris and Terry on type trees of stable graphs and prove a new variant of the Sauer-Shelah lemma for the op-rank setting of Guingona and Hill.
This work is joint with James Freitag.
April 19, 2018
Hunter Chase :
4 p.m. in SEO 427
April 26, 2018
Thomas Dean :
4 p.m. in SEO 427
Abstract
Given $n$ and $m$, it is a classical problem to find their product. In this talk, we will discuss results such as, if $n\geq 5$, then $2^n \not\leq n^2$. We will also discuss how to cancel; i.e., that $2n = 2m$ implies $n=m$. Time permitting we will discuss the more difficult cancelling by 3.
May 3, 2018
Jake Herndon :
4 p.m. in SEO 427
Aug. 30, 2018
Hunter Chase, Noah Schoem :
4 p.m. in 427 SEO
Sept. 6, 2018
Matt DeVilbiss :
4 p.m. in 427 SEO
Sept. 13, 2018
Hunter Chase :
4 p.m. in 427 SEO
Abstract
We discuss the classification of first-order theories, with an emphasis on local combinatorial properties.
Sept. 20, 2018
Noah Schoem :
4 p.m. in 427 SEO
Sept. 27, 2018
Thomas Dean :
4 p.m. in 427 SEO
Oct. 4, 2018
Matt DeVilbiss :
4 p.m. in 427 SEO
Abstract
In this talk, I will present on the basics of geometric stability theory including an introduction to combinatorial geometry. Time permitting, I will also state some of the important results of this program.
Oct. 11, 2018
Kevin Zhou :
4 p.m. in 427 SEO
Abstract
Abstract Elementary Classes (AECs) were introduced by Shelah in 1987 as a generalization of elementary classes of first-order theories. Many interesting objects of study in model theory form nonelementary classes (e.g. Archimedean ordered fields, locally finite groups, etc.), but can be expressed in some stronger logic (e.g. infintary logics such as $L_{\omega_1,\omega}$, higher order logics, etc.); AECs provide a framework to study these classes in a more general setting. In this talk, I will present the basics of AECs, as well as some important results and open problems in the classification theory of AECs.
Oct. 18, 2018
Noah Schoem :
4 p.m. in 427 SEO
Abstract
In most functional analysis courses, $\ell^1$ is often an example of a
Banach space that is not reflexive; that is, $(\ell^1)^*$, the dual space of $\ell^1$, is $\ell^\infty$,
but $(\ell^\infty)^*\supsetneq \ell^1$.
However, this argument requires the Axiom of Choice.
In fact, we will show that under a certain anti-choice axiom, that $(\ell^\infty)^*=\ell_1$.
Nov. 1, 2018
Paolo Degiorgi :
4 p.m. in 427 SEO
Abstract
Probably something related to stable theories.
Nov. 8, 2018
Thomas Dean :
4 p.m. in 427 SEO
Abstract
We define Kurepa Trees and discuss its relationship to other trees often studied in set theory.
Nov. 15, 2018
Will Adkisson :
4 p.m. in 427 SEO
Nov. 29, 2018
Hunter Chase :
4 p.m. in 427 SEO
Abstract
We explore notions of independence in model theory. We describe the development of forking and behavior of forking in different classes of theories.
Dec. 6, 2018
Alexander Berenbeim :
4 p.m. in 427 SEO
Jan. 17, 2019
Hunter Chase and Noah Schoem :
4 p.m. in 427 SEO
Jan. 22, 2019
Paolo Degiorgi :
3:30 p.m. in 427 SEO
Jan. 31, 2019
Paolo Degiorgi :
4 p.m. in 427 SEO
Feb. 14, 2019
Matt DeVilbiss :
4 p.m. in 427 SEO
Abstract
Bring a hat
Feb. 21, 2019
Thomas Dean :
4 p.m. in 427 SEO
Abstract
We define a notion of dimension on a partial order and discuss its properties. We also give a characterization of a poset having dimension 2.
Feb. 28, 2019
Hunter Chase :
4:30 p.m. in 427 SEO
Abstract
We discuss a recent paper of Simon regarding the presence of $\vee$-interpretable linear orders in unstable NIP theories.
March 7, 2019
Kevin Zhou :
4 p.m. in 427 SEO
Abstract
We review the basics of Abstract Elementary Classes (AECs) and prove some elementary results about the amalgamation property in AECs. Time permitting, we will present and prove an interesting result about a weakened version of the amalgamation property.
March 14, 2019
Noah Schoem :
4 p.m. in 427 SEO
Abstract
Solovay's original proof of the Stationary Splitting Theorem
used high-powered tools combining techniques from forcing and the study of elementary embeddings.
Though more elementary proofs are now known, the original techniques gave rise
to the study of certain nice ideals and the generic elementary embeddings they induce.
We will discuss these niceness conditions and recent work extending an old result
of Baumgartner and Taylor separating two of these niceness conditions, saturation and presaturation.
March 21, 2019
Will Adkisson :
4 p.m. in 427 SEO
Abstract
In the 16th century, the rise of Calvanism and its doctrine of predestination presented an idealogical challenge to the Catholic Church. To prevent further schisms, the Vatican was forced to seriously examine the theological consequences of a lack of free will. If all is predestined, that would imply that a good, just, and merciful God actively wills evil, rather than merely permits it, and actively predestines individuals to hell. A somewhat smaller complication that the Vatican was forced to grapple with, however, was how to describe the size of infinite sets without being able to guarantee explicit bijections to the ordinals.
April 4, 2019
Matthew DeVilbiss :
4 p.m. in 427 SEO
April 11, 2019
Noah Schoem :
4 p.m. in 427 SEO
April 18, 2019
Alexander Berenbeim :
4 p.m. in 427 SEO
Abstract
The Mekler construction constructs a 2-nilpotent group of exponent p>2 from a nice graph interpreting a structure M satisfying a theory T in a finite relational language. It is known that this group preserves stability properties like NIP and NTP_2. This talk will consist of two parts. The first part will describe this construction, along with relevant definitions, and the second part will show that this construction also preserves k-dependence.
April 25, 2019
Hunter Chase :
4 p.m. in 427 SEO
Abstract
Despite its power in (infinite) model theory, first-order logic is largely in adequate when dealing with finite structures. It is incapable of capturing relatively simple properties, such as graph connectivity. We discuss other systems of logic and their ability to capture various complexity classes.
May 2, 2019
Thomas Dean :
4 p.m. in 427 SEO
Abstract
TBA
Aug. 29, 2019
Matthew DeVilbiss and Noah Schoem :
4 p.m. in 427 SEO
Sept. 5, 2019
Kevin Zhou :
4 p.m. in 427 SEO
Abstract
We will give an overview of the basics of algorithmic randomness, with a view towards understanding some recent theorems of Reimann and Slaman. Some familiarity with basic computability theory will be helpful but not required.
Sept. 11, 2019
Matthew DeVilbiss :
4 p.m. in 427 SEO
Sept. 18, 2019
Will Adkisson :
4 p.m. in 427 SEO
Sept. 25, 2019
Paolo Degiorgi :
4 p.m. in 427 SEO
Abstract
'The proposition "P is unprovable" has a different sense afterwards - from before it was proved.
If it is proved, then it is the terminal pattern in the proof of unprovability. - If it is unproved, then what is to count as a criterion of its truth is not yet clear, and - we can say - its sense is still veiled' (RFM Part I, Appendix III, paragraph 16) .
We will look at several passages from Wittgenstein's main work on (what one could call) the philosophy of mathematics. This has been subject to much controversy.
We will probably discuss his comments on Gödel's first incompleteness theorem, and how they relate to the discussion of the role of surprise in mathematics.
While doing this, we might run into some funny wood - sellers.
Oct. 16, 2019
Thomas Dean :
4 p.m. in 427 SEO
Oct. 23, 2019
Noah Schoem :
4 p.m. in 427 SEO
Abstract
Collapsing cardinals is fairly easy,
but making regular cardinals into singular cardinals (without collapse) is more difficult.
In addition to some results from inner model theory that give lower bounds on the needed large cardinal strength,
we will also exposit on Prikry and Magidor forcing, posets of optimal large cardinal strength that make singularization possible.
Time permitting, we will also discuss applications to relevant problems in set theory.
Oct. 30, 2019
Kevin Zhou :
4 p.m. in 427 SEO
Abstract
Unfortunately, this talk will not be about model-theoretic fanfic of Harry Potter and the Order of the Phoenix. Instead, we will discuss ultrafilters and ultrapowers in model theory, and introduce Keisler's order on theories.
Nov. 6, 2019
Will Adkisson :
4 p.m. in 427 SEO
Abstract
A brief survey of some of the stronger large cardinal properties, building to a discussion of Reinhardt cardinals and Kunen's theorem.
Nov. 13, 2019
Roland Walker :
4 p.m. in 427 SEO
Abstract
Recursive saturation and $S$-saturation are interesting concepts at the intersection of model theory and recursion theory. In this talk, we will define and discuss recursively saturated models, Scott sets, effectively perfect theories, standard systems, and $S$-saturated models. We will prove that any recursively saturated model of an effectively perfect theory is $S$-saturated for exactly one Scott set $S$. It is natural to ask if all Scott sets arise in this fashion. In 1982, Knight and Nadel showed that all Scott sets of size at most $\aleph_1$ arise as the standard system of some nonstandard model of Peano Arithmetic. We will prove a more general version of this result which applies to any computable theory. This closes the question under CH. We will discuss recent progress which closes the question for specific theories when CH is not assumed.
Nov. 20, 2019
Alexander Berenbeim :
4 p.m. in 427 SEO
Abstract
This talk will be a broad survey of the combinatorial games, and a specific subclass of Partizan games known as the Surreal
numbers. The inductive construction of these games will be covered, as well as a brief survey of some key genetic functions
and relations which endow these classes with universal embedding properties. Specifically, Jacob Lurie showed that the class
of Partizan games is the universal embedding object of partially ordered abelian groups, and Philip Ehrlich, Lou van den Dries
and others have shown that the Surreal numbers are similarly an universal embedding objects for ordered groups, ordered
rings, real closed fields, logarithmic-exponential power series, and more.
Dec. 4, 2019
Noah Schoem :
4 p.m. in 427 SEO
Abstract
We can say that a set $S\subseteq\kappa$ is stationary if for any $\lambda>\kappa$
and every model $\mathcal{U}=\langle H_\lambda,\in,\dots\rangle$
there is an $M\prec \mathcal{U}$ such that $\sup(M\cap\kappa)\in S$.
But what if we want this result for a sequence of stationary sets simultaneously,
that is, given $\langle S_\alpha\mid \alpha<\tau\rangle$, each
$S_\alpha$ stationary in some $\kappa_\alpha<\tau$,
for every $\mathcal{U}=\langle H_\lambda,\in,\dots\rangle$ with $\lambda<\kappa_\tau$,
is there an $M\prec \mathcal{U}$ such that for all $\alpha<\tau$, $\sup(M\cap \kappa_\alpha)\in S_\alpha$?
We will explore what originally motivated this question and consistency results surrounding mutual stationarity.
Jan. 15, 2020
Matt DeVilbiss :
4 p.m. in 427 SEO
Jan. 22, 2020
Matt DeVilbiss :
4 p.m. in 427 SEO
Jan. 30, 2020
Noah Schoem :
3 p.m. in 427 SEO
Abstract
If $V$ is a universe of $\mathrm{ZFC}$,
one may approximate a new real, a Cohen generic over $V$,
using finite binary sequences and genericity.
However, there are other kinds of ways to add real numbers to a universe.
After a brief review of forcing and genericity, we will explore random real forcing,
Sacks forcing, and others if time permits, and discuss key properties and differences.
Feb. 6, 2020
Noah Schoem :
3 p.m. in 427 SEO
Abstract
The collection of real numbers is not fixed by the axioms of set theory;
with forcing, it is possible to add new real numbers.
After a brief recap of results from last time, we will investigate and discuss
further forcings to add real numbers and their properties.
No knowledge from the prior talk will be assumed.
Feb. 13, 2020
Kevin Zhou :
3 p.m. in 427 SEO
Abstract
Shelah introduced the notion of simple theories in 1980 as an extension of stable theories. We examine some of the motivations for studying simple theories, and explore some of the basic results about simplicity.
Feb. 20, 2020
Will Adkisson :
3 p.m. in 427 SEO
Abstract
In a previous talk, Cardinals Without Choice, I discussed the reaction of the Vatican to the doctrine of predestination as advocated by Calvinism. In the midst of this theological crisis, however, another far more earthly blight was afflicting the Catholic church: an obesity epidemic, which swept through the upper ranks of the clergy. In this talk, I will discuss the consequences of this widespread obesity through a number of lenses. The major health risks presented to high-ranking members of the church threatened a collapse of the Vatican's intricate political structure, as bishops and cardinals began to succumb to heart disease at abnormally high rates. Perhaps more importantly, it also presented a moral challenge to the legitimacy of the church. Overweight priests during times of famine and lean harvests provided a tangible symbol for the Protestant claims of Catholic decadence and hypocrisy, which threatened to undermine the moral leadership of the church.
Feb. 27, 2020
Matt DeVilbiss :
3 p.m. in 427 SEO
Abstract
TBA
March 5, 2020
Kevin Zhou :
3 p.m. in 427 SEO
Abstract
In this talk, we continue our discussion of simple theories, focusing on forking independence. We will also discuss a characterization of simple theories via abstract independence relations. No background from the previous talk will be assumed.
March 12, 2020
Paolo Degiorgi :
3 p.m. in 427 SEO
Aug. 27, 2020
No one :
4 p.m. in Zoom
Abstract
Meeting details will be e-mailed out on the LH_LOGIC listhost.
If you wish to join, please e-mail the co-organizers Noah (nschoe4 at uic dot edu)
or Matt (mdevil2 at uic dot edu) for the meeting details.
Sept. 3, 2020
Noah Schoem :
4 p.m. in Zoom
Sept. 10, 2020
Will Adkisson :
4 p.m. in Zoom
Abstract
Not to be confused with small large cardinals.
Sept. 17, 2020
Kevin Zhou :
4 p.m. in Zoom
Abstract
Model theorists discover ̶ e̶a̶r̶t̶h̶ set-shattering property of theories! More on this, and more, coming up on LHLS logic news at 4.
Oct. 15, 2020
Will Adkisson :
4 p.m. in Zoom
Abstract
Set-Theoretic Blackjack is a game for two or more players. The rules are simple: each player independently defines a new large cardinal hypothesis. Each player in turn presents their large cardinal, while the other players attempt to prove that it is inconsistent with ZFC. The player with the strongest large cardinal hypothesis that was not proven to be inconsistent is the winner.
Variant Rule: Choiceless Blackjack. The rules are the same, except that to go bust, a player's cardinal must be proven inconsistent with ZF, rather than ZFC.
Jan. 14, 2021
Matt DeVilbiss & Noah Schoem :
4 p.m. in Zoom
Jan. 21, 2021
Group :
4 p.m. in Zoom
Abstract
We will discuss the introduction and section 1 of Ben Yaacov and Usyvatsov, CONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY.
The paper can be found here:
http://math.univ-lyon1.fr/~begnac/articles/cfo.pdf
Feb. 4, 2021
Group reading :
4 p.m. in Zoom
Aug. 26, 2021
:
4 p.m. in Zoom
Abstract
We will schedule speakers and decide the format for the semester.
Sept. 2, 2021
Will Adkisson :
4 p.m. in 636 SEO
Abstract
Experts have noticed a troubling trend in the feeding patterns of the Northern Cardinal (cardinalis cardinalis). While cardinals are able to sustain a healthy diet in the wild, the overabundance of bird-feeders in suburban areas coupled with warming temperatures have led to an imbalance in their nutritional intake. This imbalance, due to a higher proportion of seeds' high lipid and caloric content, results in an increase in the average size and weight of cardinals in these areas. These large cardinals are less able to escape predators or forage on their own. In this talk we will discuss the ecological consequences of this trend, as well as some proposed solutions. No ornithological knowledge will be required, but the basics of set theory may be helpful.
Sept. 9, 2021
Kevin Zhou :
4 p.m. in 636 SEO
Abstract
Model theory is a dish originating from the relatively obscure Logic culinary tradition. For the most part, only specialists seem interested in preparing it, which is a shame because model theory not only has a rich and complex flavor, but is also very versatile and finds itself at home among many different types of mathematical meals. In this talk, we will present three possible preparations that will hopefully shed some light on this mysterious dish: an algebraic appetizer, a main-course platter of "pure" model theory, and a combinatorial custard for dessert. Note: Kevin is not a licensed chef, nor will any actual food be provided.
Oct. 7, 2021
Will Adkisson :
4 p.m. in 636 SEO
Abstract
"We live on a placid island of ignorance in the midst of black seas of infinity, and it was not meant that we should voyage far."
-HP Lovecraft, Call of Chthulu
"O horror, horror, horror! Tongue nor heart
Cannot conceive nor name thee!"
-Macbeth, Act 2 Scene 3
Oct. 21, 2021
Kevin Zhou :
4 p.m. in 636 SEO
Abstract
You've heard of Topology without Tears, now get ready for...
Nov. 4, 2021
Fanxin Wu :
4 p.m. in 636 SEO
Abstract
We discuss a result by Shelah that the fundamental group of a space that is "nice" enough must be either finitely generated or uncountable (actually of size continuum). Here "nice" means a compact metric space that is path connected and locally path connected. We follow Pawlikowski's proof which only uses some basic descriptive set theory.
Dec. 2, 2021
Fanxin Wu :
4 p.m. in 636 SEO
Abstract
We discuss the equivalence between synthetic (aka axiomatic) geometry and analytic (aka coordinate) geometry, proved by Hilbert using segment calculus, and some related results. The main step is to construct a field out of an abstract model of geometry. The construction is mostly algebraic and has little to do with logic.
Jan. 27, 2022
:
4 p.m. in 636 SEO
Abstract
We continue reading through chapter 4 of Kunen, pages 248-251.
"Forcing is what gives a set theorist his power. It's an energy field created by all living things. It surrounds us and penetrates us. It binds the galaxy together." --Obi-Wan Kenobi
Feb. 3, 2022
:
4 p.m. in 636 SEO
Abstract
We continue reading through chapter 4 of Kunen.
In stage magic, a force is a method of controlling a choice made by a spectator during a trick. Some forces are performed physically using sleight of hand, such as a trick where a spectator appears to select a random card from a deck but is instead handed a known card by the magician. Other forces use equivocation (or "the magician's choice") to create the illusion of a free decision in a situation where all choices lead to the same outcome.
Feb. 10, 2022
:
4 p.m. in 636 SEO
Abstract
We continue reading through Chapter 4 of Kunen.
"A set theorist is a human that (successfully) claims the monopoly of the legitimate use of forcing within a given territory." - Max Weber, Set Theory as a Vocation
Feb. 17, 2022
:
4 p.m. in 636 SEO
Abstract
We conclude Kunen's discussion of forcing.
Newton's Laws of Set-Theoretic Forcing:
Law 1. A set continues in its state of rest, or in uniform motion in a straight line, unless acted upon by a forcing poset.
Law 2. A set acted upon by a forcing poset moves in such a manner that the time rate of change of momentum equals the forcing.
Law 3. If two sets exert forcings on each other, these forcings are equal in magnitude and opposite in direction.
Feb. 24, 2022
Fanxin Wu :
4 p.m. in 636 SEO
Abstract
A short proof of consistency of $V\neq L$ goes like this: take a random subset $A$ of $\omega$. The probability of $A=X$ for any fixed $X\subseteq\omega$ is zero, and thus $A$ is different from all $X$, so it is new, and it cannot be in $L$ by absoluteness. We explain how this "proof" is made rigorous by the Boolean algebra formulation of forcing.
March 3, 2022
Will Adkisson :
4 p.m. in 636 SEO
Abstract
This will be a perfectly ordinary combinatorics talk, discussing Ramsey's Theorem. There is no chance that anything in this talk will be independent of ZFC. It is ludicrous to think that large cardinals might make an appearance. Frankly, I'm insulted that you would think such a thing.
March 10, 2022
Isabella Scott :
4 p.m. in 636 SEO
Abstract
Model theoretic forcing was developed in response to Cohen's proof of the independence of CH. I'll discuss some different formulations of it as well as its uses in model theory and, time permitting, computability theory.
March 17, 2022
Fanxin Wu :
4 p.m. in 636 SEO
Abstract
This will be a perfectly ordinary talk on two simple problems from calculus: (i) exponential tower; (ii) the inverse function of $(\log x)(\log\log x)$. There is no chance that these have anything to do with model theory!
March 31, 2022
Kevin Zhou :
4 p.m. in 636 SEO
Abstract
Unlike several previous LHLS talks, I am making the promise there will be (an epsilon amount of) logic content in this talk. This will be a 40-minute practice talk for the ASL North American meeting next week.
Aug. 25, 2022
:
4 p.m. in 427 SEO
Sept. 12, 2022
Will Adkisson :
5 p.m. in 427 SEO
Abstract
Cardinals tend to nest in low thickly covered foliage, and favor saplings or the lower branches of taller trees. They have been regularly spotted in mulberry, hawthorn, and dogwood trees throughout the eastern Unites States. While nests are usually built in the lower branches of these trees, male cardinals often sing while perched high in the upper branches, to allow their birdsong to carry further. This talk will discuss how the structure of trees relates to cardinal behavior, describing which kinds of trees best appeal to different types of cardinals.
Sept. 19, 2022
Carl Tang :
5 p.m. in 427 SEO
Abstract
In 1997, Hamkins constructed a tower of varying heights. We will try to gaze at the peculiar shifting tower through the widest window in SEO 427.
Sept. 26, 2022
Yutong Duan :
5 p.m. in 427 SEO
Abstract
May give an easier way to prove the stable regularity lemma according to the paper of Maryanthe and Anand of the same name of the talk.
Oct. 3, 2022
Henry Klatt :
5 p.m. in 427 SEO
Abstract
Few stories in the history of mathematics are as dramatic and entertaining as the story of the infinitesimal. Sitting at the center of the historical development of calculus, the topic pits philosophy, intuition, rigor, and computation against one another, along side a rich cast of characters including Isaac Newton, Karl Weierstrass, and Imre Lakatos. In the 1960s, Abraham Robinson placed the infinitesimal system, by then usurped by the more parsimonious Weierstrassian system of limits based on inequalities, on a rigorous footing using the techniques of model theory, creating a completely equivalent and more algebraically flavored approach to calculus. In this talk, we will review the ultrapower construction of the Hyperreals, how the system is used for calculus, and give a brief overview of the system's cultural status.
Oct. 10, 2022
Will Adkisson :
5 p.m. in 427 SEO
Abstract
They aren't large cardinals, but if you view them from the right angle and squint a bit it sort of seems like they are. Not to be confused with small large cardinals, which are totally different.
Oct. 31, 2022
Kevin Zhou :
5 p.m. in 427 SEO
Abstract
We will talk about Keisler measures, which unsurprisingly are measures that were introduced by Keisler.
Nov. 7, 2022
Ryan Carpenter :
5 p.m. in 427 SEO
Abstract
In this talk, I will give the motivation behind as well as a brief exploration into relevant logics as non-classical alternative deduction systems. After discussing relevant logics, I will talk about the work of mathematician, philosopher, and logician Robert Meyer—specifically his relevant system for arithmetic R#. We discover some surprising features of setting up mathematical axioms relevantly, and we talk about the fate of Meyer’s program in relevant arithmetic.
Nov. 21, 2022
Kay Thompson :
5 p.m. in 427 SEO
Abstract
We develop the necessary categorical notions to describe an elementary topos and relevant examples, such as categories of sets, bundles, and sheaves. We then examine how taking sheaves over a partial order relates to Cohen’s method of forcing and use this to construct a topos which ‘models’ ZFC+$\neg$CH.
Nov. 28, 2022
Isabella Scott :
5 p.m. in 427 SEO
Abstract
Abstract: Revisiting Kronecker's claim that "God created the integers, all else is the work of man", let's take a look at just how much of mathematics can be formalised in arithmetic. Since a lot of mathematics requires forming sets, we'll allow ourselves some set existence axioms, but we'll be suspicious of them, because, well, we're not sure if they have divine approval. In the interest of time, we won't talk about all of mathematics, but we'll focus on a couple of theorems about the existence of ideals in commutative rings.
Jan. 19, 2023
:
4 p.m. in 427 SEO
Jan. 30, 2023
Kevin Zhou :
4 p.m. in 612 SEO
Abstract
But why male models? This classical question was originally posed by Derek Zoolander in 2001. We do not directly address this question, but instead turn our attention to a related one involving the weighted sums of particular classes of finite models. Since weight is a sensitive topic for many, we will need to use some specialized techniques in order to handle this subject in an appropriate way.
Feb. 6, 2023
Will Adkisson :
4 p.m. in 612 SEO
Abstract
The phrase "as above, so below" is a common principle among practitioners of the occult. It first appeared in Latin in the <i>Emerald Tablet</i>, and has been repeated many times in mystical texts since. It notably appeared in the Hermetic text <i>The Kybalion</i>, believed to have been written by the American occultist William Atkinson.
Since set theory is no less arcane and mysterious than these works, it is perhaps unsurprising that the principle also appears in the study of the infinite.
Feb. 27, 2023
Bonghun Lee :
4 p.m. in 612 SEO
March 6, 2023
Ryan Carpenter :
4 p.m. in 612 SEO
Abstract
In this talk, I will introduce the system of modal model theory, an augmented style of first-order logic with remarkable expressive power and desirable model-theoretic properties. We will focus on one particular system of modal model theory: modal graph theory. As it turns out, modal graph theory interprets a very large portion of set-theoretic truth; we investigate whether modal graph theory can interpret full set-theoretic truth, and therefore whether it may serve as the foundation for mathematics.
March 13, 2023
Gregoire Fournier :
4 p.m. in 612 SEO
Abstract
Recent advances in AI have raised concerns about its black-box nature and
lack of explainability. We follow a line of research that aims at characterizing AI through a finite model theoretic point of view. In particular, we study the graph neural network architecture (GNN), its proximity with the Weisfeiler-Lehman (WL) class of algorithms, and graded modal logic.
April 17, 2023
Kay Thompson :
5 p.m. in 427 SEO
Abstract
In this talk, we will generalize the Fraïssé construction to arbitrary regular cardinals and arbitrary categories. We explore some classical examples and some applications of the generalized form.
Aug. 31, 2023
:
4 p.m. in 427 SEO
Sept. 7, 2023
Will Adkisson :
4 p.m. in 427 SEO
Abstract
Square principles are one of the most prevalent examples of an anti-compactness property in set theory, appearing in many different contexts. Using them as our guide, we will embark on a tour of combinatorial set theory, introducing a number of different properties and describing they interact with square. In particular we will discuss stationary reflection, Godel's constructible universe L, various large cardinals, and the tree property. We will also mention how to obtain square via forcing.
Sept. 14, 2023
Kevin Zhou :
4 p.m. in 427 SEO
Abstract
We'll discuss some of the recent connections between theoretical machine learning and mathematical logic.
Sept. 28, 2023
Carl Tang :
4 p.m. in 427 SEO
Abstract
We take a short walk through the concepts required to understand the statement of Sela's solution to Tarski's problem. Then, if time permits, we walk around the hall of model-theoretic results following Sela's solution.
Oct. 5, 2023
Ryan Carpenter :
4 p.m. in 427 SEO
Abstract
This sentence is false; or is it? In this talk, I will motivate a rather provincial viewpoint in the philosophy of mathematics (using Gödel's first incompleteness theorem) which embraces the existence of paradoxes, and I will detail one mathematician's attempt to take this viewpoint seriously. In doing so, we will discuss Cantor's theorem in both classical and paraconsistent settings, we'll investigate relevant set theory, and we'll do a bit of inconsistent reasoning along the way.
Oct. 12, 2023
Michael Lange :
4 p.m. in 427 SEO
Abstract
The Omitting Types Theorem for countable first-order theories has, in addition to a model-theoretic proof, a "purely" topological proof making use of the Baire Category Theorem. This proof will be given, and then the notion of omitting types will be discussed for a more general notion of type space abstracted from the Stone space of types over a first-order theory.
Oct. 19, 2023
Bonghun Lee :
4 p.m. in 427 SEO
Abstract
TBA
Nov. 9, 2023
David Gonzalez :
4 p.m. in 427 SEO
Abstract
Abstract: The concept of Scott sentence complexity was introduced by Alvir, Greenberg, Harrison-Trainor and Turetsky and gives a way of assigning countable structures to elements of the Borel hierarchy. By calculating the Scott sentence complexities occurring in a class of structures we obtain a detailed picture of the descriptive complexity of its isomorphism relation. We study possible Scott sentence complexities of linear orderings using two approaches. First, we investigate the effect of the Friedman-Stanley embedding on Scott sentence complexity and show that it only preserves complexities. We then take a more direct approach and exhibit linear orderings of all Scott sentence complexities except and for a limit ordinal. We show that the former can not be the Scott sentence complexity of a linear ordering. In the process we develop new techniques which appear to be helpful to calculate the Scott sentence complexities of structures in general.
This talk is based on joint work with Dino Rossegger.
Nov. 16, 2023
Will Adkisson :
4 p.m. in 427 SEO
Abstract
Since $\aleph_1$ is the second infinite cardinal, it should be obvious and non-controversial that it is not in any way a large cardinal. Nonetheless, in this talk we will justify this clearly self-evident fact. We will list a number of combinatorial properties of large cardinals, including some which can consistently hold at small cardinals, and demonstrate that $\aleph_1$ does not satisfy them.
Jan. 11, 2024
:
4 p.m. in 427 SEO
Jan. 18, 2024
Kevin Zhou :
3 p.m. in 427 SEO
Abstract
We'll discuss a connection between the learning-theoretic topic of sample compression and the model-theoretic topic of definability of types, and some relatively recent results in this area.
Jan. 25, 2024
Will Adkisson :
3 p.m. in 427 SEO
Abstract
We will discuss the systems of logic that best capture the behavior of the set-theoretic multiverse. Will modal logic make an appearance? Possibly.
Feb. 1, 2024
Carl Tang :
3 p.m. in 427 SEO
Abstract
We will go through what we know so far about elementary equivalences of groups (mainly finitely generated).
Feb. 22, 2024
Yutong Duan :
3 p.m. in 427 SEO
Abstract
I may quickly go over several ways to prove strong minimality and show the proof of "D_3 implies stongly minimal".
March 7, 2024
Ryan Carpenter :
3 p.m. in 427 SEO
Abstract
In this talk, I will introduce hyperreal numbers both axiomatically and through an ultraproduct construction. We will examine the structure of the hyperreal numbers, do a bit of nonstandard analysis, and consider "the" hyperreals as it interacts with ZFC.
March 14, 2024
Kevin Zhou :
3 p.m. in 427 SEO
Abstract
No, holding up a spork and calling yourself "t3h PeNgU1N oF d00m" will not do it. Relevant xkcd: https://xkcd.com/1210/. A book of a million random digits may also make a guest appearance.
March 28, 2024
Michael Lange :
3 p.m. in 427 SEO
Abstract
This talk will introduce some basic definitions and facts about infinitary logic. The main focus will be on Scott's Isomorphism Theorem for languages allowing finite quantification and countable conjunction.
April 4, 2024
Kevin Zhou :
3 p.m. in 427 SEO
Abstract
We'll discuss a recent paper of Malliaris and Moran exploring parallels between the finitary and infinitary in stability theory, algorithms, and machine learning.
April 11, 2024
Will Adkisson :
3 p.m. in 427 SEO
Abstract
In this talk we will discuss a variety of ways to destroy cardinals, and the built-in safety measures these techniques use to keep collateral damage to a minimum. Then we will remove those safety measures, and show how to harness the resulting chaos for our own ends.
April 18, 2024
Ryan Carpenter :
3 p.m. in 427 SEO
Abstract
To make model theoretic arguments more efficient, one often defers to properties of the so-called "monster model" to pull back desired results to the model within which one works. This convenient tool can seem mysterious and suspect—oftentimes, it makes use of class-sized objects—but it need not. The goal of this talk is to make clearer in the speaker's mind what exactly the monster model is and why it is a legitimate tool in model theoretic arguments; perhaps the audience will find some of the remarks useful as well. Along the way, we will review the concepts of saturation, homogeneity, transcendence (of a theory), and stability.
Sept. 4, 2024
Yutong Duan :
2 p.m. in 427 SEO
Abstract
I would like to introduce the theory of Differential closed field (DCF), and some paper results about strongly minimal equations.
Sept. 11, 2024
Ryan Carpenter :
2 p.m. in 427 SEO
Abstract
I will attempt to succinctly present a modified version of Gerhard Gentzen's proof of the consistency of PA. This approach is adopted from course notes by Henry Towsner, a contemporary proof-theoricist, so the presentation matches what current research efforts in proof theory look like. In the talk, I will detail a few deduction calculi, prove results on the elimination of certain of their rules for special theories, and show how these results lead toward a consistency proof for Peano Arithmetic. To close, we will consider the proof of the main result in light of Godel's theorems.
Sept. 18, 2024
Ryan Carpenter :
2 p.m. in 427 SEO
Abstract
In part one of this talk, we introduced the Tait-style deduction calculus for PA, proved some elementary results, and showed how a proof of CUT-elimination for PA would entail its consistency. In this second part of the talk, we will define an auxiliary theory and proceed through a four-step procedure toward the end of showing that PA admits CUT-elimination for existential sequents, hence establishing its consistency à la Gentzen.
Sept. 25, 2024
Rishi Banerjee :
2 p.m. in 427 SEO
Abstract
Descriptive set theorists study the properties of “definable” (i.e., Borel or analytic) subsets of “nice” (i.e., Polish) topological spaces. The study of these subsets is closely linked to the model theory of the infinitary logic $L_{\omega_1\omega}$, which extends first order logic by allowing countably infinite conjunctions and disjunctions. Of particular interest is the Borel reducibility hierarchy for Borel equivalence relations, which can be thought of as a way of comparing the complexity of classification problems. A common technique is to look at the “local structure” of an equivalence relation, i.e., what types of first order structures can be placed on all the equivalence classes in a uniform Borel manner. A countable Borel equivalence relation (CBER) is a Borel equivalence relation in which every equivalence class is countable. We show that there is an Lw1w-theory precisely axiomatizing the local structure of CBERs. Background in descriptive set theory and infinitary logic will not be required. (Talk based on joint work with Ronnie Chen, University of Michigan)
Oct. 2, 2024
Carl Tang :
2 p.m. in 427 SEO
Abstract
How much information first-order logic can detect is a natural question in logic. In this talk, I will present Thomas Koberda's proof of first-order rigidity of homeomorphism groups of compact manifolds.
Oct. 16, 2024
Bonghun Lee :
2 p.m. in 427 SEO
Oct. 23, 2024
Devrim Pekmezci :
2 p.m. in 427 SEO
Abstract
We shall establish that a subset A of a group G, which is NIP and has positive measure under a left-invariant finitely additive probability measure on its Boolean algebra of left translates, is weakly generic, while presenting the relevant concepts from the general theory as we progress through the proof. The proof will rely on tools from combinatorics and topological dynamics, which will be introduced throughout the talk. Time permitting, we will discuss some questions revolving around the main theorem.
Oct. 30, 2024
Yutong Duan :
2 p.m. in 427 SEO
Abstract
When can we keep $\mathcal{C}$-internality of a type under a definable function.
Jan. 27, 2025
Ryan Carpenter :
3 p.m. in 612 SEO
Abstract
In his 1988 paper "On Groups and Fields Definable in o-Minimal Structures", Anand Pillay gives a characterization of groups definable in o-minimal structures by which they look like manifolds over the underlying o-minimal structure. He notes that the argument runs virtually unchanged for o-minimally definable fields, and he uses this to deduce that all o-minimally definable fields are either real or algebraically closed. In this talk, I will present the relevant background material on o-minimality as well as these three key results which highlight a way by which we may consider o-minimal structures as nice.
Feb. 3, 2025
Yutong Duan :
3 p.m. in 612 SEO
Abstract
I would like to give a proof of the type-definability of a groupoid structure that can be defined when a type has relative internality. And then I may show some examples.
Feb. 17, 2025
John E. Solak :
3 p.m. in 612 SEO
Abstract
It is common Model Theory lore that topological spaces are not first order i.e. cannot be fully captured in first order Logic. More formally, we say that topologies do not form an elementary class in any (reasonable) language. Yet, determining how to prove this seems to be a challenging task.
In this talk, I will showcase various mode-theoretic results about "suitable" languages as evidence of this claim. I will assume familiarity with introductory Model Theory. Knowledge of the Compactness and Lowenheim-Skolem Theorems is recommended.
Feb. 24, 2025
Michael Lange :
3 p.m. in 612 SEO
Abstract
The goal of this talk is to provide some motivation for the definitions of forking and dividing. If time permits, we will also discuss characterizations of some model-theoretic dividing lines in terms of the behavior of forking.
March 3, 2025
Hazal Aydogdu :
3 p.m. in 612 SEO
Abstract
Fix an L-theory T and a formula. We will define what it means for a formula to be stable and explore equivalent reformulations, following Shelah’s Unstable Formula Theorem. In particular, we will study the connections between the cardinality of type spaces, n-ladders, n-trees, and the definability of types.
March 10, 2025
Rishi Banerjee :
3 p.m. in 612 SEO
March 31, 2025
Devrim Pekmezci :
3 p.m. in 612 SEO
Abstract
We have been discussing how the study of model theory is structured by drawing dividing lines based on the presence or absence of various combinatorial configurations. For example, we introduced the order property and the binary tree property to characterize stable theories. In this talk, we will focus on the independence property and introduce NIP theories—those that lack the independence property. We will also explore how NIP theories generalizes different features of stable theories we previously discussed.
April 7, 2025
Carl Tang :
3 p.m. in 612 SEO
Abstract
Set theorists Hardin and Taylor proved an interesting theorem about predicting the "future values" of a function based on its "history". We will see the proof and how this theorem connects to other ideas in set theory. Warning: this talk contains potentially fictitious investment strategies.
Sept. 10, 2025
Ryan Carpenter :
4:30 p.m. in 427 SEO
Abstract
Transcendental number theory, tropical geometry, and cohomology of complex abelian varieties—to name a few—are some of the areas from which techniques are borrowed to prove existential closedness results that are active areas of research in model theory today. Such techniques (no doubt, the questions they serve to answer) might suggest a sort of existential question for the model theorist: what am I doing in the middle of all this? In this talk, I will detail the existential closedness results model theorists have cared about over the last thirty years, and then I will place these results in their proper historical context to get around to answering the model theorist's existential question.
Oct. 8, 2025
John E. Solak :
4 p.m. in 427 SEO
Abstract
In this talk, I will provide an introduction to o-minimal theories and some of the Geometry and Topology available to those who study them. The introduction will be based heavily on portions from Lou Van den Dries’ book, <i>Tame Topology and O-minimal Structures</i>, as well as Ronnie’s reading course from last Fall. After the introduction, I will discuss the definitions of dimension and Euler characteristic in the o-minimal setting, and then survey some of their applications while bringing to light small personal concerns over the intuitiveness of these measurements.
Oct. 15, 2025
Daniel Ibaibarriaga :
4 p.m. in 427 SEO
Abstract
In this talk I will introduce the notion of a beautiful pair, which generalizes Poizat’s belle paires of stable structures. I will derive some of its basic properties and explain its relation to the strict pro-definability of the space of definable types.
Oct. 22, 2025
Yutong Duan :
4 p.m. in 427 SEO
Abstract
In this talk, I will give the idea to prove that $DCF_0$ has $2^\lambda$ non-isomorphic models for any cardinal $\lambda$.
Oct. 29, 2025
Devrim Pekmezci :
4 p.m. in 427 SEO
Abstract
Newelski introduced tools from topological dynamics into model theory to generalize the notion of a generic type from stable group theory to arbitrary first-order theories. He adapted Ellis’s theory to the definable setting and introduced what is now known as the Ellis Group Conjecture, which relates the Ellis group of the space of types concentrating on a definable group G, viewed as a G-flow, to the maximal compact quotient of G, a model-theoretic invariant, under suitable tameness assumptions. The conjecture has been established for definably amenable groups in NIP theories by Chernikov and Simon. In this talk, we will outline the conjecture’s motivation, introduce the key concepts surrounding it, and discuss possible directions for further research.
Nov. 12, 2025
David Meretzky :
4 p.m. in Zoom
Abstract
This talk will give an overview of the Picard-Vessiot differential Galois theory beginning with Liouville's theorem on integration in finite terms and ending with the contemporary state of the theory and it's generalizations. Differential Galois theory was conceived of by Sophus Lie as being a differential analogue of the theory of Galois. The main theorem of it's first iteration (work of Picard and Vessiot in the 1890s) is an analogue of the theorem on solvability by radicals. Developments in early 20th century mathematics allowed Ellis Kolchin to give a fully algebraic account of the Picard-Vessiot theory. We will survey the work of the Kolchin school and describe closely related model-theoretic perspectives and generalizations. Time permitting, we will discuss connections to other schools of differential Galois theory and applications.
Nov. 19, 2025
Karthik Ravishankar :
4 p.m. in 427 SEO
Feb. 25, 2026
Devrim Pekmezci :
3 p.m. in 512 SEO
Abstract
The notion of randomization was introduced by H. Jerome Keisler and later refined by Itaï Ben Yaacov and Keisler within the framework of continuous model theory. A randomization of a first-order structure M is a continuous structure with two sorts: one for random elements of M, and one for events in an underlying probability space. All randomizations of a fixed structure M are models of the same complete continuous theory, which admits a natural axiomatization and quantifier elimination. In this talk, we shall briefly review some background on continuous model theory and the model theory of atomless probability spaces, then define the randomization of a first-order structure, present its axiomatization, and conclude with examples and properties of randomization.
April 22, 2026
John Solak :
3 p.m. in 512 SEO
Abstract
In this 20-minute talk, I will present the historical context in which O-Minimality arose with a little timeline. After this, I will provide basic definitions and discuss many of the major theorems available to any who wish to use this tameness condition. Lastly, honing in on the tame topology and geometry available to those who employ O-Minimality, I will describe recent research of Pablo Guerrero in the direction of general O-Minimal topology.