Departmental Colloquium : Past Events
Past Seminars
The following seminars have already happened, you may instead view upcoming seminars in this series.
Oct. 21, 2005
Sheldon Katz :
3 p.m. in SEO 636
Abstract
The goal of enumerative geometry is to count geometric configurations
satisfying prescribed geometric conditions. A classical example: there
are exactly 2 lines in space meeting each of 4 lines in general position.
But many other 19th century problems remained unsolved for more than a
century.
During the last 15 years, enumerative geometry has been revitalized from
an unlikely source: string theory in physics. The ideas of string theory
gave birth to entirely new areas of geometry which have been used to
solve classical enumerative questions (and much more).
In this talk I give some perspective on enumerative geometry and explain
how string theory has given new insight about the enumeration of curves.
I also illustrate with examples and give some prospects for the future.
Oct. 28, 2005
Eric Friedlander :
3 p.m. in SEO 636
Abstract
Understanding modular representations of finite groups, $p$-restricted
Lie algebras and related structures is a central challenge in algebra.
Beginning with work of Daniel Quillen, various mathematicians have
used cohomological techniques to produce geometric structures
which reflect properties of modular represenations. In the past couple
of years, Julia Pevtsova and I have introduced elementary represenation-
theoretic techniques which refine and generalize these cohomological
invariants. In current work with Andrei Suslin, Julia and I use
these techniques to introduce invariants which are new even for very
elementary finite groups.
Nov. 4, 2005
Fred J. Hickernell :
3 p.m. in SEO 636
Abstract
Laboratory experiments, computer experiments, and numerical
algorithms all require designs, i.e., the set of points where the
input function is evaluated. Unfortunately, in practice, the design
is not often given much thought. Grids aligned to the coordinate
axes are popular, but they become too costly as the number of
variables increases. Independent and identically distributed random
points allow one to overcome the curse of dimensionality, but they
usually lead to suboptimal answers. This talk describes several
families of good designs, such as orthogonal arrays, low discrepancy
points and minimax designs. For seemingly unrelated problems it is
found that a good design spreads points out evenly. Moreover, to
avoid the curse of dimensionality, low dimensional projections of the
design must also spread points evenly. The correct measures of even
spread are typically based on a distance or a reproducing kernel
(symmetric, positive definite matrix). This talk surveys known
results and poses open problems. No specialized background is
assumed, but the speaker intends to demonstrate how a variety of
ideas in mathematics, statistics, and computer science are needed to
construct and evaluate good designs.
Nov. 11, 2005
Robert Bruner :
3 p.m. in SEO 636
Abstract
If we apply a generalized cohomology theory to the classifying
space of a group, we obtain a ring whose structure reveals
something about the group. As a first approximation we can
consider the variety defined by this ring. The first result in
this direction was the work of Quillen in 1971, describing the
variety defined by the mod p cohomology in terms of the
elementary abelian subgroups. A number of such results for
other cohomology theories have now been proven. We shall review
these and discuss in detail the case of connective K-theory,
which provides a deformation from representation theory to
cohomology (at the level of varieties) and speculate on
generalizations of these results.
Nov. 21, 2005
Martin C. Golumbic :
3 p.m. in SEO 636
Abstract
Graph theory and its applications is an exciting mathematical discipline
which motivates the search for new algorithms, exact structures and
combinatorial properties. To illustrate this point, consider the
following simple question:
When can you factor a logic (Boolean) formula, into a (logically
equivalent) form in which each variable appears once and only once?
For example, the function f = ab + acd + ace satisfies this property since
it can be factored into the "read-once" expression f = a(b+c(d+e)).
However, the function h = ab + bc + cd does not satisfy the property.
Formally, a Boolean function f is called a read-once function if it has a
(logically equivalent) factored form in which each variable appears
exactly once, called a read-once expression for f.
Read-once functions have interesting combinatorial properties and often
arise in real circuit applications. They have also gained recent interest
in the field of computational learning theory.
In this talk, we will present the mathematical and computational aspects
of this problem. We will show several classical characterizations of
read-once functions which involve combinatorics, graph theory and
properties of positive (monotone) Boolean functions. We also present the
first polynomial time algorithm for recognizing and factoring read-once
functions. The algorithm is based on a theorem of Gurvich and on algorithms
for cograph recognition and a new efficient method for checking normality.
Finally, we raise a number of questions regarding the factoring certain
non-read-once functions. In particular, we are able to show that if the
co-occurrence graph of a positive Boolean function f is a tree, then the
function is read-twice. However, no characterization is known for general
read-twice functions.
(Joint work with Aviad Mintz and Udi Rotics)
Feb. 10, 2006
Peter Olver :
3 p.m. in SEO 636
Abstract
In this talk, I will describe a new approach to the powerful Cartan
theory of moving frames that is based on equivariant maps. The method
is completely algorithmic, and can be readily applied to completely
general finite-dimensional Lie group and even infinite-dimensional
pseudo-group actions. After introducing the basic ideas, I will
attempt to survey a wide variety of new applications, including
classification of differential invariants, invariant variational
problems and differential equations, symmetries and equivalence of
polynomials in classical invariant theory, object recognition in
computer vision, and the design of symmetry-preserving numerical
approximations.
Feb. 22, 2006
Der-Chen Chang :
3 p.m. in SEO 636
Abstract
Let D be an (appropriate) domain in n-dimensional Euclidean space with
smooth boundary. In this talk, we shall discuss two questions. First,
what are the possible (natural) notions of Hardy space that generalize the
usual Hardy spaces on the whole space? For the second question consider
the Dirichlet and Neumann problems for the domain D. In the context of the
relevant Hardy spaces, can one obtain the boundedness of sharp estimates
for the Green functionsof the Dirichlet and Neumann problems?
March 3, 2006
Robert Ghrist :
3 p.m. in SEO 636
Abstract
As sensor engineering and manufacturing evolve
to produce smaller devices, we will face the problem of dealing
with large collections of local networked devices for communication,
security, and surveillance. Engineers increasingly need to solve
global problems with swarms of local sensors.
Topologists solved a similar problem nearly a century ago, using
only combinatorial data from local objects (simplices) to derive
global data (homology). Unfortunately, homological algebra has
not been very successful in crossing over to engineering. This
talk will demonstrate the surprising effectiveness of algebraic
topology as a toolbox for working with sensor networks having
neither localization capabilities nor probabilistic guarantees.
March 27, 2006
Alexei Bondal :
2 p.m. in SEO 636
Abstract
We will explain why derived categories are important in
algebraic geometry and outline recent methods and trends
in studing derived categories of coherent sheaves
on algebraic varieties.
March 31, 2006
Naomi Fisher :
3 p.m. in SEO 636
Abstract
In the 1980s mathematics education gained national prominence once again. Two
events-- the 1986 Tulane Conference on calculus reform, and the 1989 publication
of NCTM's Curriculum and Evaluation Standards for School Mathematics leading to
the development of reform K-12 curricula-- have had extensive influence on
mathematics education. In this period, too, the Mathematicians and Education
Reform (MER) Forum, then the MER Network, was organized in 1988.
The vision of founding Co-Directors, Harvey Keynes, University of Minnesota, and
Philip Wagreich, University of Illinois at Chicago, for meetings of
mathematicians engaged in pre-college education, grew to include pre-college,
undergraduate, and graduate education. MER has been a continuing presence in
the mathematics community through more than 30 national workshops and annual
special sessions on Mathematics and Education Reform, and its publications.
Viewed from the speaker's 18 years experience in MER, this talk looks at the
history of MER, and discusses lessons learned. Some questions considered are:
why is it important for mathematicians to be involved in the mathematics
education enterprise? how can mathematicians contribute effectively to
education? what has changed, since the 1980s, and what has not?
April 7, 2006
Bela Bollobas :
3:15 p.m. in SEO 636
Abstract
Percolation theory was founded by Broadbent and Hammersley in
1957, in order to model the flow of fluid in a porous medium with
randomly blocked channels. By now, the field has blossomed into a
huge area, with thousands of papers, many books, and exciting
connections to several branches of mathematics and physics.
In the talk I shall introduce and study one of the basic
parameters of percolation theory, the critical probability. I
shall sketch greatly simplified proofs of two fundamental and
classical theorems, and I shall present some recent results
obtained jointly with Oliver Riordan.
The talk should be understandable to people with a minimal
background in probability theory.
Bernd Sturmfels :
2 p.m. in SEO 636
Abstract
The hyperdeterminant of format 2 x 2 x 2 x 2 is a polynomial of degree 24
in 16 unknowns which has 2894276 terms. We compute the Newton polytope of
this polynomial and the secondary polytope of the 4-cube. The 87959448
regular triangulations of the 4-cube are classified into 25448
D-equivalence classes, one for each vertex of the Newton polytope. The
4-cube has 80876 coarsest regular subdivisions, one for each facet of the
secondary polytope, but only 268 of them come from the hyperdeterminant.
Joint work with Debbie Grier, Peter Huggins and Josephine Yu
(math.CO/0602149)
April 28, 2006
Irena Peeva :
2 p.m. in SEO 636
Abstract
The Hilbert function is an invariant which measures the
size of a graded ideal in a polynomial ring: in particular,
it gives the dimension and the multiplicity of the variety
defined by a prime ideal. This talk will focus on
the properties of ideals with a fixed Hilbert function, starting
from classical results of Hilbert, Macaulay, and Grothendieck. The
Hilbert scheme (introduced by Grothendieck) parameterizes ideals with
a fixed Hilbert function. The talk will discuss properties of Hilbert
schemes.
Sept. 29, 2006
Mark Ronan :
3 p.m. in SEO 636
Abstract
One November evening in 1978, a mathematician named John McKay was
sitting at home in Montreal reading some papers in number theory, and was
astonished to see the number 196,884. It was apparently the smallest
non-trivial coefficient for a function of great importance to number
theorists, yet in his own area of mathematics (group theory) the number
196,883 had recently appeared as the smallest possible, non-trivial
dimension for the Monster. This mysterious coincidence led to some
amazing observations, now embraced by the term moonshine, leading
from number theory, through group theory, to string theory in
mathematical physics.
The Monster itself is a finite group. All finite groups are built up from
simple groups, and the talk will start by explaining how these
simple groups came to be discovered. There are a dozen different
families, along with 26 exceptions called sporadic groups and the
largest exception is the Monster. The discovery of these exceptions is a
fascinating story, involving many mathematicians over many years, and we
shall follow it through to the moonshine mysteries inspired by McKay^Òs
observation. The talk will be non-technical, and will say as much about
the mathematicians as the mathematics.
Oct. 13, 2006
Alex Eskin :
3 p.m. in SEO 636
Abstract
The notion of quasi-isometry is the natural equivalence relation if
one views groups as metric spaces. We prove that any group
quasi-isometric to the three dimenionsional solvable Lie group Sol is
virtually a lattice in Sol. Our results extend to some other classes
of groups and spaces, and are contributions to Gromov's program for
classifying finitely generated groups up to quasi-isometry [Gr2].
We introduce a new technique for studying quasi-isometries, which we
refer to as "coarse differentiation". This is joint work with David
Fisher and Kevin Whyte.
Oct. 27, 2006
John D'Angelo :
3 p.m. in SEO 636
Abstract
CR Geometry concerns CR manifolds and CR mappings between them.
CR stands for both Cauchy-Riemann and Complex-Real.
In this talk the source and target CR manifolds will be the unit spheres
in (generally different dimensional) complex Euclidean spaces. We start
by contrasting an elementary result on the unit disk with a result of
Herb Alexander (UIC, 1974) concerning proper self-mappings between balls. This
discussion leads us to analyzing the relationship between the target dimension
and how complicated the mapping can be. By making a simplifying
assumption, we obtain a beautiful combinatorial problem asking for bounds
on the degree of a real polynomial (with nonnegative coefficients, and
constant on a hyperplane) in terms of the number of its distinct
monomials. In two dimensions Lens spaces play a key role. The talk
culminates with two recent results (obtained with Jiri Lebl and Han
Peters) obtaining various bounds on the degrees of these mappings.
Nov. 30, 2006
Izzet Coskun :
1 p.m. in SEO 636
Abstract
A Littlewood-Richardson rule is a positive rule for computing
the structure constants of the cohomology ring of flag varieties with
respect to their Schubert basis. In recent years new geometric
Littlewood-Richardson rules have led to the solution of many important
problems, including Klyachko, Knutson and Tao's solution of Horn's
conjecture and Vakil's solution of the reality of Schubert calculus. In
this talk I will survey some of the basic geometric ideas that underlie
these Littlewood-Richardson rules.
Dec. 4, 2006
Alex Gorodnik :
3 p.m. in SEO 636
Abstract
We start with a discussion of some fundamental conjectures
in arithmetic geometry that describe the set of solutions of Diophantine
equations in terms of geometric invariants of the corresponding
algebraic varieties. In particular, we mention the Batyrev-Manin
conjecture on the number of rational points and the Peyre conjecture
on the asymptotic distribution of rational points. We explain how to attack
these conjectures in the case of homogeneous varieties using
either representation theory or dynamics on homogeneous spaces.
Dec. 8, 2006
Laszlo Babai :
3 p.m. in SEO 636
Abstract
In an unusual confluence of concepts emerging from a diverse set
of areas about 15 years ago, the Abelian Sandpile Model offers
a rich structure studied by communities in statistical physics,
probability theory, algebraic combinatorics, theoretical computer
science, discrete dynamical systems.
The process under consideration takes a connected graph, puts
"grains of sand" on each "site" (node); when the "pile" at a
site gets too tall, it "topples," passing a grain to each neighbor.
One of the sites is a "sink;" it swallows all the grains that reach
it. We repeat the process until the configuration stabilizes
("avalanche"). Then we add another grain at some site and
start all over. The process was introduced in 1988 by Bak, Tang, and
Wiesenfeld as a model of the phenomenon of "self-organized criticality"
in statistical physics. The evolution of the system is a "visual feast"
(Creutz, 1991); the images show largely unexplored fractal phenomena.
The study of this diffusion process has led to the assignment of
algebraic structures to (rooted) graphs (Dhar, 1990): a commutative
monoid (the "sandpile monoid") on the set of stable configurations;
and an abelian group, the "sandpile group," on the important subset of
"recurrent configurations." The sandpile group, which is the unique
mimimal ideal of the sandpile monoid, is a particularly intriguing
algebraic invariant of the graph. It is related to the lattice
generated by the rows of the reduced Laplacian of the graph; and
from this connection it follows that its order is the number of
spanning trees of the graph, thus adding group structure to
Kirchhoff's celebrated "matrix-tree theorem" (1848).
After a general introduction to the subject along the lines of
joint work with Evelina Toumpakari, I will mention recent
work on the "transience class" of a graph, i.e., the maximum number
of grains that can be added to the empty configuration before it
becomes recurrent. This number is also the nilpotence class of a
related semigroup. While this number can be exponentially large
compared to the size of the underlying graph, we prove that for
the particularly interesting case of the square grid, it is
polynomially bounded. This result is joint work with Igor Gorodezky.
Open problems of diverse nature arise; they involve statistics,
dynamical systems, algebra, number theory, algorithms and
computational complexity.
Feb. 16, 2007
Gregory Lawler :
3 p.m. in SEO 636
Abstract
A number of lattice models in two-dimensional statistical
physics are conjectured to exhibit conformal invariance
in the scaling limit at criticality. In this talk, I will
try to explain what the previous sentence means, focusing on
three elementary examples: simple random walk, self-avoiding walk,
loop-erased random walk. I will describe the
limit objects,
Schramm-Loewner Evolution (SLE), the Brownian loop soup, and
the normalized partition functions, and show how conformal
invariance can be used to calculate quantities ("critical
exponents") for the model. I will also describe why (in some
sense) there is only a one-parameter family of conformally
invariant limits. In conformal field theory, this family
is parametrized by central charge.
Much of the talk will be based on joint work with Oded
Schramm and Wendelin Werner although I will discuss
work by a number of other researchers.
This talk is for a general mathematical audience. No knowledge
of statistical physics will be assumed.
Feb. 23, 2007
Dan Dugger :
3 p.m. in SEO 636
Abstract
A "sums-of-squares formula" is a certain type of algebraic
identity, considered by Hurwitz in his investigation into the
existence of composition algebras. One of the earliest applications
of algebraic topology was to prove that certain kinds of sums-of-squares
formulas cannot exist over the real numbers, or more generally over
fields of characteristic zero. The question of what happens over
characteristic p fields was raised in the 1970s, but then left
unsettled as algebraic techniques proved inadequate. In this talk
I'll describe some recent progress on this problem using etale
homotopy theory, and I'll try to explain how the problem ties in
to current work in motivic homotopy theory.
March 2, 2007
Richard A. Brualdi :
3 p.m. in SEO 636
Abstract
I will discuss certain ideas related to the diagonals and diagonal
structure of a square matrix, with connections to the polytope of doubly
stochastic matrices, its simplex faces, ray-nonsingularity, determinantal
regions, digraphs, and bipartite graphs.
March 9, 2007
Lou Kauffman :
3 p.m. in SEO 636
Abstract
In the 1960's Roger Penrose discovered a remarkably lucid and
fundamentally topological diagrammatic method for handling the mathematics
of quantum particles with spin a multiple of 1/2. Calling this method
"spin networks" he proposed to reconstruct space-time from
a world of spin-exchange processes "pior" to space and time.
Penrose proved a Spin-Geometry Theorem that
reconstructed directions in a three-dimensional space from the abstract
networks. SpaceTime remained a problem for the theory.
In the 1980's the speaker discovered a topological generalization
of the Penrose spin networks that included a new knot invariant, the
Jones polynomnial. One can construct these q-deformed spin nets
on purely topological grounds, and they form models for topological
quantum field theories and specific models for the colored Jones
polynomials and the Witten-Reshetikhin Invariants of three manifolds.
The q-deformed spin networks form a bridge between combinatorial and
quantum field theoretic approaches to these invariants.
In the last few years Michael Freedman, Alexei Kitaev and their co-workers
showed that quantum computing can, in principle, be universally performed
within a
topological quantum field theory. This means that certain topological
quantum field theories have a rich enough structure to support unitary
representations of the braid group that are dense in the unitary groups.
A quantum computer is a unitary transformation that can be applied to a
prepared quantum state in such a way that measuring the result of the
transformation yields computational or algorithmic information.
In the 1990's Peter Shor showd that quantum computer's can factorize
integers faster than classical algorithms.
This talk will explain how the q-deformed spin nets and the bracket
model of the Jones polynomial provide a simple and accessible
construction of unitary braid representations that are universal for
quantum computation. We will begin with the Penrose spin nets, then
generalize them to include the Jones polynomial, then construct
the promised representations. We will then discuss how these representations
are related to the dream of physically realized quantum computing via the
possibilities inherent in the physics of two-dimensional media.
March 16, 2007
Liviu Nicolaescu :
3 p.m. in SEO 636
Abstract
Two Morse functions on a smooth manifold M are called equivalent
if we can obtain one from the other by global changes of coordinates on M
and R (the real numbers). V.I. Arnold has computed the number of such equivalence
classes when M=S^1, and formulated a precise conjecture concerning the
asymptotics of these numbers when M= S^2.
I will explain how to describe the generating function of the numbers of
Morse functions on S^2 in terms of certain elliptic integrals, and then
how to use this to prove Arnold's conjectured asymptotics.
March 23, 2007
Benny Sudakov :
3 p.m. in SEO 636
Abstract
An (n,d,lambda)-graph is a d-regular graph on n vertices so that the
absolute value of each eigenvalue of its adjacency matrix, besides the
largest one, is at most lambda. I will survey some of the remarkable
pseudo-random properties of such graphs in which lambda is much smaller
than d, describe various constructions, and present several applications
of these graphs in the solution of problems in Extremal Combinatorics,
Geometry and Complexity.
April 13, 2007
Charles Fefferman :
3 p.m. in SEO 636
Abstract
The abstract is available at
http://www2.math.uic.edu/~lewis/scv2007/abstracts/abstracts.pdf#fefferman
Sept. 14, 2007
Dave Marker :
3 p.m. in SEO 636
Abstract
The definable sets in the real field
with exponentiation have many of the good topological and geometric
properties of semialgebraic sets. If one looks at the complex
numbers the situation is much more complicated as you can
define the integers. Yet, there is some hope for an interesting model
theory. I will discuss an abstract approach to complex exponentiation due
to Zilber and some interesting number theoretic and geometric problem
it raises.
Oct. 4, 2007
Mircea Mustata :
3 p.m. in SEO 636
Abstract
In characteristic zero, one can define invariants of singularities
via valuations, or equivalently, via resolutions of singularities. In
positive characteristic, invariants can be defined in a very elementary way,
using
the Frobenius homomorphism. This talk will be about analogies and
connections between these two points of view (with focus on positive
characteristic).
Oct. 12, 2007
Kevin Pilgrim :
3 p.m. in SEO 636
Abstract
Classical, Riemannian, expanding conformal dynamical systems on the
Riemann sphere include convex cocompact Kleinian groups and
hyperbolic rational maps. These systems have lots of
generalizations in lots of different directions. I'll discuss some
of them and, along the way, point out connections to dynamics,
algebra, coarse geometry, and analysis on metric spaces.
Oct. 15, 2007
Navin Singhi :
4 p.m. in SEO 636
Abstract
A new class of ternary rings -- semiadditive rings -- is defined. A free
semiadditive ring plays a similar role in studying planar ternary rings as
the ring of integers or polynomial rings over integers play in studying
fields. Methods developed recently to study free semiadditive rings will
be discussed. A well known result of G. Higman on loops and its
generalization to semiadditive rings will be discussed.
Two well known conjectures on projective planes state that the order of
every finite projective plane is a power of a prime number and that a
finite projective plane with no proper subplanes is isomorphic to a plane
coordinatized by a prime field. Application of the above mentioned
results to these problems will be discussed.
Oct. 18, 2007
Nicholas Jackiw :
3:30 p.m. in 1000 SEO
Abstract
In considering the geometric figure, Kant distinguishes between image -- the
traditional visual diagram -- and schemata, the generalized concept of that
diagram that "never exist anywhere except in thought." Dynamic Geometry figures
produced by software such as The Geometer's Sketchpad bridge this divide through
flexible, "rubbery" diagrams that (under manipulation) transform into any valid
realization of their defining geometric constraints, but at any instant retain
the immediacy and tangibility of specific images. In this talk, Sketchpad's
author explores the implications of Dynamic Geometry visualization on
mathematical inquiry and pedagogic practice in the context of school (6-12)
mathematics, and describes implementation concerns in the design and development
of Dynamic Geometry software.
Oct. 19, 2007
Nathalie Sinclair :
1 p.m. in 1007 W. Harrison Street, Room 2019 BSB
Abstract
In this talk, I seek to establish insight into varieties of modeling that occur,
recurrently, in students', teachers', and curriculum developers' experiences
with The Geometer's Sketchpad. In this process, I attempt to contrast the
conventional, practical sense of mathematical modeling -- modeling of situations
and phenomena to generate and predict plausible outcomes -- to at least two
other available forms or types of modeling practice, that I find especially
relevant to mathematics education in their foundational didactic intent.
Oct. 26, 2007
Frank Sottile :
3 p.m. in SEO 636
Abstract
Understanding the real solutions to systems of polynomial equations is a difficult question with many applications.
In particular, a non-trivial lower bound is an existence proof for solutions and non-trivial upper bounds give complexity bounds.
The best-known upper bound is due to Khovanskii and is unrealistically large.
Lower bounds are a very recent phenomenon, having arisen with real Gromov-Witten invariants and (separately) with the
Wronski map in Schubert calculus. The results, though, are
striking: "Most" rational curves of degree d interpolating
3d-1 real points in the plane are real, and every rational
function with only real critical points is real.
In this talk, I will describe this background and then
discuss recent work giving upper and lower bounds on the
numbers of real solutions to some sparse polynomial systems.
This is joint work with Soprunova (lower bounds) and with Bihan and Bates (upper bounds).
Nov. 9, 2007
Joe Harris :
3 p.m. in SEO 636
Abstract
Faltings' theorem that there are only finitely many
solutions to a diophantine equation of genus 2 or more has naturally
led to speculation about possible extensions: how the theorem might
be generalized to higher dimensions, and how the number of solutions
behaves when we vary the coefficients of the problem, to name two. In
this talk we'll discuss some of these generalizations, and describe a
surprising logical connection between them.
Nov. 16, 2007
Bun Wong :
3 p.m. in SEO 636
Abstract
Abstract: In this talk I will describe certain
rigidity
phenomenon and a problem on the compact complex manifolds with ample
cotangent bundles. Their connection with an old question of Hirzebruch
and other open problems on smooth four manifolds theory will be discussed.
Nov. 19, 2007
Michael Mandell :
3 p.m. in SEO 636
Abstract
This series of talks explores the relationship between algebraic
K-theory and "abstract homotopy theory", or the theory of mapping
spaces. The first talk is designed to be accessible to graduate
students. It discusses some of the motivation and background as well
as some current results and open problems. The remaining talks
discuss related topics in more detail.
Nov. 30, 2007
Dorian Goldfeld :
3 p.m. in SEO 636
Abstract
A Dirichlet series is an infinite series of type a(1) + a
(2)2^{-s} + a(3) 3^{-s} + ... where s is a complex variable.
Examples include the Riemann zeta function. A multiple Dirichlet
series (MDS) is a Dirichlet series in a variable s whose
coefficients a(1), a(2), ... are Dirichlet series in another
variable, etc. MDS may be viewed as Dirichlet series in several
complex variables. We survey what is known about MDS and explain
connections of MDS with the theory of classical Lie groups.
Jan. 14, 2008
Hyman Bass & Mark Thames :
3 p.m. in SEO 636
Abstract
Improved mathematics achievement requires changes in the instructional dynamics in classrooms and in the mathematical education of teachers. Unfortunately, while there is broad agreement on the need for better mathematical preparation of teachers, there is a lack of clarity about what mathematics teachers need to know. To address this problem, we examine actual practice. We define mathematical knowledge for teaching (MKT) as the mathematical knowledge, skills and habits of mind entailed in the work of teaching and we study it empirically. One result has been the identification of a set of mathematical practices also key to teaching and learning. Surprisingly prominent has been the use of definitions. In this presentation, we introduce the notion of MKT and investigate the role of definitions in teaching and learning. In particular, we illustrate where and how definitions arise in the work of teaching.
Feb. 8, 2008
Izzet Coskun :
3 p.m. in SEO 636
Abstract
The interpolation problem asks "Is there a polynomial of
degree $d$ with prescribed Taylor polynomials at given points?"
This problem comes up in contexts ranging from algebraic geometry
to statistics. I will describe cases where we know the solution
of the problem---focusing mainly on the geometric
aspects. I will also explain the connections between the interpolation
problem and other famous problems such as the problem of finding the
dimension of secant varieties.
Feb. 13, 2008
Michael Singer :
3 p.m. in SEO 636
March 7, 2008
Winnie Li :
3 p.m. in Lecture Center D-2
Abstract
Unlike congruence modular forms, the arithmetic of
noncongruence modular forms is not much understood, due to
the lack of Hecke operators. For noncongruence forms, Atkin and
Swinnerton-Dyer proposed a substitute of the familiar degree two recursive
relation satisfied by congruence Hecke eigenforms by three-term
congruence relations. In a certain situation, this yields very interesting
congruence relations between the Fourier coefficients of congruence and
noncongruence forms.
In this survey talk, we shall review the development of noncongruence
forms, discuss the progress on congruence relations, as well as the
unbounded denominator conjecture, which asserts that the algebraic
noncongruence forms are distinguished by its Fourier coefficients having
unbounded denominators. We shall see that the modularity of certain Galois
representations plays an essential role.
March 20, 2008
Bernard Deconinck :
3 p.m. in SEO 636
Abstract
Going back to considerations of Benjamin (1974), there has been
significant interest in the question of stability for the stationary
periodic solutions of the Korteweg-deVries equation, the so-called
cnoidal waves. In this paper, we exploit the squared-eigenfunction
connection between the linear stability problem and the Lax pair for
the Korteweg-deVries equation to completely determine the spectrum
of the linear stability problem for eigenfunctions that are bounded
on the real line. We find that this spectrum is confined to the
imaginary axis, leading to the conclusion of spectral stability. An
additional completeness argument allows for a statement of linear
stability.
April 10, 2008
Song-Ying Li :
3 p.m. in SEO 636
Abstract
In this talk, I will talk about the nature and applications of the
complex Monge-Amp\`ere operators in complex analysis. I also connect the
Monge-Amp\`ere operator to
pseudo-Hermitian structures, pseudo Ricci curvature, pseudo scalar
curvatures of CR manifolds. Finally, I will provide some applications of
Monge-Amp\`ere operators for characterizations of balls.
April 11, 2008
Robin Hartshorne :
3 p.m. in Lecture Center D4
Abstract
Noether's theorem says that a surface of degree at least four in
projective three-space contains only complete intersection curves. I
will
review the background and history of this result, and then show a
modern
proof using deformation theory.
April 18, 2008
Richard Schwartz :
3 p.m. in SEO 636
Abstract
Outer billiards is a basic dynamical system defined
relative to a convex shape in the plane. In many
cases, the system produces beautiful and mysterious
tilings of the plane that are not well understood,
and also the dynamics of these systems are not well
understood. I will survey some results in this subject,
paying particular attention (of course) to my own.
Aug. 25, 2008
Marshall Hampton :
3 p.m. in SEO 636
Abstract
The study of solutions of polynomial systems is a fundamental
topic in mathematics. In this talk I will discuss the finiteness and
enumeration of solutions to families of polynomial systems that arise in
celestial mechanics and vortex dynamics - however, the methods used can be
applied in greater generality. I will also briefly introduce the Sage
computational platform which is used for this work. Sage is free,
open-source, and builds on many other successful projects. For this talk,
I will focus on the components Gfan, PHCpack, Singular, and some
Sage-native code.
Sept. 19, 2008
Josephine Yu :
3 p.m. in SEO 636
Abstract
We apply tropical geometry to study the image of a map defined
by Laurent polynomials with generic coefficients. If this
image is a hypersurface then our approach gives a construction
of the Newton polytope of its defining equation. This construction
can be used to compute mixed fiber polytopes, including secondary
polytopes. No knowledge of tropical geometry will be assumed,
and the talk will include an introduction to this subject.
This is based on joint works with Bernd Sturmfels and Jenia Tevelev.
Sept. 26, 2008
Shanyu Ji :
3 p.m. in SEO 636
Abstract
We start to review Cartan's moving frame
theory for real submanifolds in Euclidean space,
for complex submanifolds in the Complex
Projective space, and in particular for CR
submanifolds in the Heisenberg Space which is
bimeromorphic to a sphere. Then we use this
theory to study holomorphic proper mappings
between balls.
Oct. 3, 2008
János Kollár :
3 p.m. in SEO 636
Abstract
The simplest Cremona transformation of projective 3-space
is the involution
$\sigma:(x_0:x_1:x_2:x_3)\mapsto
\left(\frac1{x_0}:\frac1{x_1}:\frac1{x_2}:\frac1{x_3}\right),$
which is a homeomorphism outside the "coordinate tetrahedron"
$(x_0x_1x_2x_3=0)$.
By studying the action of $\sigma$ on real quadric surfaces,
we show that $\sigma$ and its conjugates generate a dense subgroup
of $Homeo(S^2)$, the group of homeomorphisms
of the 2-sphere.
Then we show that the same holds if the 2-sphere is replaced
by the torus or by any non-orientable surface
and explain why there can not be similar results for
orientable surfaces of genus $\geq 2$.
(Joint work with Frédéric Mangolte.)
Oct. 13, 2008
Volker Mehrmann :
2 p.m. in SEO 636
Abstract
In the last 30 years, differential-algebraic equations (DAEs) have become one
of the standard modeling tools in many application areas, like
electronic circuit simulation, multibody dynamics, chemical
engineering, control theory, etc. The reason is that via DAEs the
dynamics of the system can be modeled componentwise and
different components can be linked via interfaces or constraints.
This advantage in modeling on the other hand puts much harder
requirements on the analysis and numerical methods. For these
reasons DAEs have been studied in detail in the last 40 years
with respect to their dynamical behavior, existence and uniqueness
of solutions, numerical methods and control.
Oct. 24, 2008
Guershon Harel :
3 p.m. in SEO 636
Abstract
Most students, even those who desire to succeed in school, are intellectually aimless in mathematics classes because often they do not realize an intellectual need for what we intend to teach them. The notion of intellectual need is inextricably linked to the notion of epistemological justification: the learners' discernment of how and why a particular piece of knowledge came to be. This talk addresses historical and philosophical aspects of these two notions, as well as ways teachers can be aware of students' intellectual need and address it directly in the mathematics classroom.
Oct. 31, 2008
Dan Mauldin :
3 p.m. in SEO 636
Abstract
The following problem has its origins in some early work of Ulam
and Oxtoby in the foundations of ergodic theory. For each r, 0< r <1, let
m(r) be the measure on the Cantor space induced by an infinite sequence of
independent Bernoulli trials. The question is: given m(r) and m(s), when
is there a homeomorphism, h, of Cantor space such that m(r)(E) =
m(s)(h(E)), for each Borel set E. We will answer this question and will
pose a number of related still unsolved questions
Nov. 7, 2008
Laura Matusevich :
3 p.m. in SEO 636
Abstract
The systematic study of hypergeometric functions and differential
equations in one variable was originated by Euler and Gauss over two
hundred years ago. The higher dimensional case also has a distinguished
history. For instance, it is a theorem of Mellin that the roots of a
polynomial are hypergeometric functions of its coefficients. It has been
noted in the last 20 years, following ideas of Gelfand, Graev, Kapranov
and Zelevinsky, that tools from algebraic geometry are relevant and useful
in the study of hypergeometric equations; the goal of this talk is to
illustrate this phenomenon. I will mention joint results with Christine
Berkesch, Alicia Dickenstein, Ezra Miller, Timur Sadykov and Uli Walther.
Nov. 20, 2008
Brad Efron :
2 p.m. in LC A1
Abstract
Familiar statistical estimates such as batting averages, political polls, and medical trial results are obtained by direct observation of cases of
interest. Sometimes, though, we can learn from the experience of "others": for instance there may be information about Player A's
batting ability in the observed averages of Players B,C,D,... I will present several examples showing how this
works in practice, indicating some of the surprising theoretical ideas involved. The talk is mainly descriptive in nature, and is
intended for a general scientific audience.
Nov. 21, 2008
Alexander Razborov :
3 p.m. in SEO 636
Abstract
The concepts of proof and computation are central to virtually
any intellectual human activity. However, before the remarkable
advances of the 20th century mathematical logic, the border between
them had never been sharp even in the context of pure mathematics. It
was only in the work of these great logicians that the separation
between proofs and computations became rigorous and (as many
thought) final.
This talk is devoted to a *reverse* trend that results from adding the
new element of feasibility (or efficiency) into this millennial-old
brew. Strangely enough, this seemingly ``orthogonal'' addition has lead to
many new intriguing facets of the interaction between (slightly
compromised versions of) proofs and computations not existing in the
classical world. We will attempt a highly informal tour through
the areas where these new phenomena happen, like Propositional Proof
Complexity, NP-Completeness and (time permitting) Interactive and
Probabilistically Checkable Proofs. And we will try to convey, using
concrete and intertwined examples, the general feeling of complexity
and intricacy of relations between proofs and computations in this
emerging reality.
Jan. 14, 2009
Hyman Bass & Mark Thames :
3 p.m. in SEO 636
Abstract
Improved mathematics achievement requires changes in the instructional dynamics in classrooms and in the mathematical education of teachers. Unfortunately, while there is broad agreement on the need for better mathematical preparation of teachers, there is a lack of clarity about what mathematics teachers need to know. To address this problem, we examine actual practice. We define mathematical knowledge for teaching (MKT) as the mathematical knowledge, skills and habits of mind entailed in the work of teaching and we study it empirically. One result has been the identification of a set of mathematical practices also key to teaching and learning. Surprisingly prominent has been the use of definitions. In this presentation, we introduce the notion of MKT and investigate the role of definitions in teaching and learning. In particular, we illustrate where and how definitions arise in the work of teaching.
Feb. 6, 2009
Alan Frieze :
3 p.m. in SEO 636
Abstract
Random graphs have been an object of research since the pioneering work
of Erdos and Renyi in the 1960's. We will present a survey of results
and open problems, old and new.
Feb. 13, 2009
Michael Singer :
3 p.m. in SEO 636
Abstract
I will develop a Galois theory of linear difference
equations where the Galois group are linear differential groups
that is, groups of matrices whose entries satisfy a fixed set of
polynomial differential equations. These groups measure the
differential dependence among solutions of linear difference
equations. I will show how this theory can be used to reprove
Hoelder's Theorem that the Gamma function satisfies no
differential polynomial equation as well as new results
concerning differential dependence of solutions of higher order
difference equations.
Feb. 20, 2009
Alex Iosevich :
3 p.m. in SEO 636
Abstract
We shall explore several different manifestations of the following general principle which says that if a subset of a vector space is large, in a suitable sense, than it contains a rigid copy of every finite geometric configuration. In the continuous setting, the size is typically measured using the Hausdorff dimension. In the discrete setting, the cardinality of the set plays this role. In vector spaces over finite fields, a combination of the two phenomena comes into play. An interplay of combinatorial, number theoretic and Fourier analytic methods is involved and the relationship between the different manifestations of the configuration problems will be emphasized throughout.
March 6, 2009
Charlie Doering :
3 p.m. in SEO 636
Abstract
It is still not known whether solutions to the 3D Navier-Stokes
equations for incompressible flows in a finite periodic box can
become singular in finite time. (Indeed, this question is the subject
of one of the 1M Clay Prize problems.) It is known that a solution
remains smooth as long as the enstrophy, i.e., the mean-square vorticity,
of the solution is finite. The generation rate of enstrophy is given by
a functional that can be bounded using elementary functional estimates.
Those estimates establish short-time regularity but do not rule out
finite-time singularities in the solutions. In this work we formulate
and solve the variational problem for the maximal growth rate of
enstrophy and display flows that generate enstrophy at the greatest
possible rate. Implications for questions of regularity or singularity
in solutions of the 3D Navier-Stokes equations are discussed. This
is joint work with Lu Lu, Indiana University Mathematics
Journal Vol. 57, pp. 2693-2727 (2008).
March 13, 2009
Ted Chinburg :
3 p.m. in SEO 636
Abstract
I'll begin this talk by describing
some famous finiteness theorems and conjectures.
These were motivated in part trying to carry
out certain kinds of calculations, e.g. by trying
to describe explicitly all the rational points on an elliptic
curve. I'll then describe a new finiteness problem which
arises from studying deformations of complexes
of modules for Galois groups.
March 20, 2009
John Smillie :
3 p.m. in SEO 636
Abstract
Low complexity and high complexity dynamical systems
differ from one another in fundamental ways. I will discuss some
of the properties of low complexity systems and some of the special
techniques which can be used to understand them in the situation
that arises in polygonal billiards.
April 3, 2009
Bettina Pedemonte :
3 p.m. in SEO 636
Abstract
Several research works in the domain of algebra highlight that many algebraic learning difficulties are
rooted in the passage from arithmetic to algebra and in the different way mathematical expressions and
propositions (equations and inequations) are used in these domains. As a matter of fact, a cognitive gap
emerges when students pass from operating with numerical expressions and propositions composed only
by numbers that are typical of arithmetic, to literal expressions and propositions, composed also by letters
that characterize algebra.
These works have oriented the design and the accomplishment of a new system named ALNUSET (ALgebra
on the NUmerical SETs) to support the teaching and learning processes in the algebraic domain. ALNUSET
was developed in the context of ReMath (IST - 4 - 26751) EC project and is designed for students of lower
and upper secondary school (yrs 12-13 to 16/17). It is constituted by three integrated components: the
Algebraic Line component, the Algebraic Manipulator component, and the Cartesian plan component.
This seminar focuses on the influence of ALNUSET on students' construction of proof. Results of an
experiment are presented to show in which way it can be used to enhance the teaching and learning of algebraic proof.
References:
Pedemonte B., Chiappini G. (2008) ALNUSET: a system for teaching and learning algebra International Journal of
Continuing Engineering Education and Life-Long Learning (IJCEELL) 18 5/6 Inderscience Publishers, 627 - 639.
Pedemonte B. The algebraic manipulator of Alnuset: a tool to prove Proceeding of the Sixth Congress of the
European Society for Research in Mathematics Education CERME 6, Lyon, Francia.
April 10, 2009
William McCallum :
3 p.m. in SEO 636
Abstract
There has been much debate about how to teach high school
mathematics in recent years, and some discussion of changing the mix
of topics in the high school curriculum. However, there areas of the
curriculum where the topics themselves seem to be archeological
relics, encrusted with years of neglect. Are they trash or treasure?
Perhaps a systematic program of excavation will help us reconstruct a
coherent architecture of high school mathematics. In this talk I will
look at a few examples of areas of the curriculum that could bear
deeper examination.
April 17, 2009
Bernd Sturmfels :
3 p.m. in SEO 636
Abstract
Convex algebraic geometry is an emerging field at the interface
of convex optimization and algebraic geometry. A primary focus lies on the
geometric underpinnings of semidefinite programming. This lecture offers a
self-contained introduction. Starting with elementary questions concerning
multifocal ellipses in the plane, we move on to discuss singularities and
projections of spectrahedra, and new algorithms for real algebraic varieties.
April 24, 2009
Benjamin Miller :
3 p.m. in SEO 636
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.
Sept. 4, 2009
Rahul Pandharipande :
3 p.m. in SEO 636
Abstract
I will discuss old and new points of view on the
cohomology of the basic moduli spaces in
algebraic geometry: Grassmannians, map spaces,
and curves. The talk will start with ideas of Schubert,
Mumford, and Faber and end with some new results.
Sept. 11, 2009
Jerzy Filar :
3 p.m. in SEO 636
Abstract
Solutions of many of most important mathematical problems depend
on values of one or more parameters. Sometimes, values of these
parameters are known rather precisely, sometimes they are only
rough estimates of true values and, interestingly, there are also
important situations where previously ``missing" parameters can be
inserted into a problem to aid its analysis. Furthermore, there
are often situations when configurations of two or more parameters
arise where the nature of solutions to the original problem
changes qualitatively at these precise <i>critical
configurations</i>. This presentation outlines two, seemingly
separate, lines of research that are nonetheless connected by the
underlying approach that focuses on the importance of
understanding the behaviour of solutions to problems in
neighbourhoods of critical values of parameters.
The first line considers the classical (NP-hard) problem of
determining whether a given graph possesses a Hamiltonian cycle.
Exploiting singularly perturbed controlled Markov chains and their
fundamental matrices we reduce the problem to that of minimising
the variability of the ``first return time" to the home node. The
latter is a highly structured, non-convex, optimisation problem
whose properties shed light on both the theoretical complexity of
the problem and its algorithmic tractability. A recent innovation
in this approach is the restriction to doubly stochastic Markov
chains.
The second line considers a perturbed mathematical programming
problem where both the objective and the constraint functions are
polynomial in all underlying decision variables and in the
perturbation parameter $\varepsilon.$ We study the behaviour of
the solutions of such a perturbed problem as $\varepsilon
\rightarrow 0.$ Though the solutions of mathematical programming
problems are real, we consider the Kuhn-Tucker optimality system
as a $1-$dimensional complex algebraic variety in a
multi-dimensional complex space. We then use Buchberger's
elimination algorithm of the Gröbner bases theory to replace the
defining equations of the variety by its Gröbner basis, that has
the property that one of its elements is bivariate, that is, a
polynomial in $\varepsilon$ and only one of the decision
variables.
We conclude by speculating that the above algebraic reduction
procedure lends itself to the identification of <i>critical
parameter configurations</i> in a wide range of mathematical models
describing interactions between natural and human development
processes.
Sept. 18, 2009
Jeong Han Kim :
3 p.m. in SEO 636
Abstract
We consider the problem of finding an unknown graph by using two types of queries with an additive property. Given a graph, an additive query asks the number of edges in a set of vertices while a cross-additive query asks the number of edges crossing between two disjoint sets of vertices. The queries ask sum of weights for
the weighted graphs. These types of queries were partially motivated in DNA shotgun sequencing and linkage discovery problem of artificial intelligence. For a given unknown weighted graph $G$ with $n$ vertices, $m$
edges, and a certain mild condition on weights, we prove that there exists a non-adaptive algorithm to find the edges of $G$ using $O(m \log{n}/\log{m})$ queries of both types provided that $m \geq n^{\epsilon}$ for any constant $\epsilon> 0$. For a graph, it is shown that the same bound holds for all range of $m$.
This settles a conjecture of Grebinski for finding an unweighted graph using additive queries. We also consider the problem of finding the Fourier coefficients of a certain class of pseudo-Boolean functions. A similar coin weighing problem is also considered.
(Joint work with S. Choi)
Oct. 2, 2009
William Notz :
3 p.m. in SEO 636
Abstract
In this talk I will discuss two settings in which I have applied adaptive experimental designs for computer experiments. The first involves adaptive (sequential) designs for deciding where to observe the computer code in order to produce a statistical predictor with good overall fit to the code. The second involves adaptive designs for estimating quantiles of the induced distribution on the output of the computer code given distributions for the inputs to the code. If time permits, I will mention a third setting in which sequential designs are used to decide where to observe the computer code for purposes of calibration (deciding what values of certain tuning parameters produce the best agreement between the code and actual data from the physical process the code simulates).
Oct. 9, 2009
Bruce Berndt :
3 p.m. in SEO 636
Abstract
In the spring of 1976, while searching through papers of the late
G. N. Watson at Trinity College, Cambridge, George Andrews found a
sheaf of 138 pages in the handwriting of Srinivasa Ramanujan,
generally regarded as India's greatest mathematician. In view of
the fame of Ramanujan's earlier notebooks, Andrews naturally
called these papers Ramanujan's "lost notebook." This work,
comprising about 650 results with no proofs, arises from the last
year of Ramanujan's life and represents some of his deepest work.
First, we provide a history of the lost notebook. Second, a
general description of the topics found in the lost notebook will
be provided. For some of the topics, such as ranks and cranks of
partitions, we offer some details. The third portion of the
lecture will be devoted to a more detailed discussion of one of
the topics prominently addressed in the lost notebook, namely
continued fractions.
Oct. 16, 2009
William Fulton :
3 p.m. in Lecture Center D4
Abstract
In this expository talk, we show how a Riemann-Roch formalism leads to a
simple proof of a general formula for restrictions of equivariant line
bundles to fixed points. On homogeneous varieties it gives Weyl's
character formula, and on toric varieties it gives Brion's formula for
lattice points in polytopes. This is based on ideas of George Quart in the
1970's and recent conversations with Bill Graham.
Nov. 6, 2009
Bryna Kra :
3 p.m. in SEO 636
Abstract
The connection between ergodic theory and additive
combinatorics dates back to the 1970's, with Furstenberg's proof of
Szemeredi's Theorem via ergodic theory. Certain algebraic constraints
(arising from nilsystems) play a key role in understanding objects
that arise in Furstenberg's proof. More recently, nil-structures have
been imported into the finite combinatorial setting, playing a role in
finding patterns in the primes. I will give an overview of where
nilsystems arise in ergodic theory and topological dynamics,
explaining some of the connections to additive combinatorics.
Nov. 9, 2009
Kevin C. Moore :
3 p.m. in SEO 636
Abstract
Understanding and using trigonometric functions is difficult for both students and secondary teachers. These difficulties range from weak understandings of topics foundational to trigonometry (e.g., angle measure and function) to incoherent conceptions of the various contexts in which trigonometry is applied (e.g., the unit circle and right triangles). As an example, students often have difficulty reasoning about trigonometric functions as functions defined on the real numbers. This talk reports results of an investigation into the understandings and reasoning abilities involved in learning ideas of trigonometry. The data was collected in the context of a teaching experiment designed to support precalculus students in developing conceptions of angle measure, images of the radian as a unit of measurement, and connections across the contexts of trigonometry. It was hypothesized that these foundational conceptions constructed by the students would support the students in developing coherent understandings of trigonometric functions. The curriculum also promoted student reasoning abilities (e.g, quantitative and covariational reasoning) and function understandings that are foundational for learning central ideas of calculus. Findings from the investigation revealed information about student understandings of angle measure that are needed to understand and use trigonometric functions. Specifically, the study gained insight to the role of student conceptions of the radian as a unit of measurement when students are asked to reason about angle measure and trigonometric functions. Analysis of the collected data also illuminated the critical role of students' conceptualization of quantities as varying, prior to formalizing an understanding of sine and cosine as functions defining the relationship between two covarying quantities.
Nov. 13, 2009
Lazslo Lempert :
3 p.m. in SEO 636
Abstract
One of the great realizations of twentieth century
mathematics was that there is a huge variety of problems
out there whose solvability can, and should, be
packaged in terms of cohomology groups. The talk
will be about cohomology groups that arise when
trying to solve certain fundamental analytical
problems of complex analysis and geometry.
After introducing the central notion of the
subject: holomorphic functions, I will pass to
the so called Cousin problem, and its significance.
I will summarize the works of H. Cartan, Oka, and
Serre on the solution of the Cousin problem in
finite dimensional spaces. The solution is expressed
in terms of cohomology groups. Finally, I will
discuss recent developments in infinite dimensions.
Jan. 11, 2010
Benjamin Miller :
3 p.m. in SEO 636
Abstract
Since the inception of the subject, dichotomy
theorems have played a central role in the development
of descriptive set theory. We will discuss approaches to
establishing such theorems using ideas from ergodic
theory and graph theory.
Jan. 15, 2010
Christof Sparber :
3 p.m. in SEO 636
Abstract
We shall explain how nonlinear dispersive equations of Schrodinger type can be used to effectively
describe the collective dynamics of (large) quantum systems. After a short review of the basic analytical
properties of the nonlinear equations obtained, we shall discuss recent mathematical results for some
extended models, arising in the description of Bose-Einstein condensates. Time permitting, we shall also
discuss some asymptotic results in multi-scale regimes.
Feb. 12, 2010
Christian Rosendal :
3 p.m. in SEO 636
Feb. 19, 2010
Ryan Martin :
3 p.m. in SEO 636
Abstract
Mixture distributions make for useful statistical models, but
estimation of the mixing distribution itself is computationally and
theoretically challenging. Recent progress has been made with a fast,
online algorithm called predictive recursion (PR), which is capable of
producing a continuous estimate of the mixing density. While the
computational strengths of PR are readily apparent, good theoretical
properties have been much slower to develop. I will present a new and
general theorem which says that, if the mixture model is mis-specified,
then the PR estimate of the mixture density converges almost surely to
the Kullback-Leibler projection of the true density onto the model,
provided that the kernel satisfies a certain uniform
square-integrability condition. From this, almost sure weak convergence
of the PR estimate of the mixing distribution follows as a corollary.
The fact that PR is a special case of the Robbins-Monro stochastic
approximation process will be used to sketch a proof of the main
theorem. I will also give a bound on the rate of convergence, and
discuss its minimax nature. Some extensions, applications, and open
questions will also be considered.
Feb. 22, 2010
Martin C. Golumbic :
3 p.m. in SEO 636
Feb. 26, 2010
Jeffrey Brock :
3 p.m. in SEO 636
Abstract
Given a closed surface S of negative Euler-characteristic, a self-homeomorphism of S that is not isotopic to the identity is called "pseudo-Anosov" if it preserves no family of simple closed curves on S up to isotopy. Due to Thurston, the notion of a pseudo-Anosov homeomorphism plays a dual role in Teichmüller theory, where it induces a hyperbolic isometry of the Weil-Petersson metric on Teichmüller space, and in the study of 3-dimensional manifolds, where it serves as the monodromy for a surface bundle over the circle with a complete hyperbolic structure.
There is a fixed constant K depending only on the Euler characteristic of S so that for a given pseudo-Anosov f:S->S the volume of the corresponding surface bundle M_f and the translation length of f on Teichmüller space have ratio lying in the interval [1/K,K]. This relationship motivates questions on nature and depth of the connection between these two invariants of the homeomorphism f.
We suggest and develop some new connections between the collection of volumes of surface bundles and the Weil-Petersson translation distances of their pseudo-Anosov monodromy maps, while introducing a fundamental confounding disconnect between these two quantities in general, answering a question of Manin and Marcolli.
March 5, 2010
Michael Wolf :
3 p.m. in SEO 636
Abstract
We prove that there is only one way to 'desingularize' the intersection of two planes in space to and obtain a periodic minimal surface as a result. The proof is mostly an exercise in, and an introduction to, basic Teichmuller theory: we translate the geometry of minimal surface in space into a statement about a moduli space of flat structures on Riemann surfaces, and then study deformation theory and degenerations in this moduli space to prove the result. We remark on the general (non-periodic) case.
March 8, 2010
JM Landsberg :
3 p.m. in SEO 636
Abstract
L. Valiant conjectured an algebraic variant
of problem to compare the complexity classes P and
NP, where one instead compares the determinant and permanent
polynomials. K. Mulmuley and M. Sohoni have proposed a program
to prove Valiant's conjecture using geometry and
representation theory, which they call the
Geometric Complexity Theory (GCT) program. I will give
an overview of the GCT program, and describe recent work
on the GCT program with L. Manivel and N. Ressayre which led us to
solve a classical problem in algebraic geometry regarding
dual varieties.
March 19, 2010
Mihnea Popa :
3 p.m. in SEO 636
Abstract
I will describe a recently emerged parallel between classical
geometry in projective space and not so classical geometry on abelian
varieties. I will state some conjectures and a few results, and will aim
most of the talk to relative beginners.
April 2, 2010
Ravi Vakil :
3 p.m. in SEO 636
Abstract
Four ordered points on the projective line, up to projective
equivalence, are classified by the cross ratio, a notion introduced by
Cayley. This theory can be extended to more points, leading to one of
the first important examples of an invariant theory problem, studied
by Kempe, Hilbert, and others. Instead of the cross ratio (a point on
the projective line), we get a point in a larger projective space, and
the equations necessarily satisfied by such points exhibit classical
combinatorial and geometric structure. For example, the case of six
points is intimately connected to the outer automorphism of $S_6$. We
extend this picture to an arbitrary number of points, completely
describing the equations of the moduli space. This is joint work with
Ben Howard, John Millson, and Andrew Snowden. This talk is intended
for a general mathematical audience, and much of the talk will be
spent discussing the problem, and an elementary graphical means of
understanding it.
April 9, 2010
Sergey Yuzvinsky :
3 p.m. in SEO 636
Abstract
The idea to view the cohomology ring A^* of a space as a cochain complex with
the differential d_a (a \in A^1 ) given by the multiplication by a appeared first in
the Farber-Novikov spectral sequence. The first sheet of the sequence is formed
by cohomology H^* (A^* , d_a ). For a compact space X the sequence converges to the
cohomology of X with local coefficients determined by a. Independently the cochain
complex has been studied for complex hyperplane arrangement compelments which
led to the notion of resonance varieties.
These varieties will constitute the main character of the talk. Recently they
have been related to classical geometric structures on the complex projective plane,
pencils of algebraic curves, and also the Bernstein-Sato polynomials. If time allows,
we will briefly discuss some generalizations of properties of the resonance varieties
to other classes of spaces.
April 16, 2010
Mihai Paun :
3 p.m. in SEO 636
Abstract
We will recall some basic facts concerning Bergman kernels and explain their relevance in the context of semi-positivity of relative canonical bundles.
April 23, 2010
William Goldman :
3 p.m. in SEO 636
April 30, 2010
Stephen Kudla :
3 p.m. in SEO 636
Abstract
The locally symmetric varieties $Y=\Gamma\backslash D$ where $D$ is the unit ball in $\mathbb C^n$ and $\Gamma$ is an arithmetic
subgroup of $U(n,1)$ have a rich geometry and arithmetic. Such quotients have many `special' algebraic cycles
arising as quotients $Z_x = \Gamma_x\backslash D_x$ of sub-balls $D_x \subset D$. By results of old joint work with
John Millson, the generating series for the cohomology classes determined by suitable collections of such cycles
are modular forms for unitary groups $U(r,r)$. Recently, in joint work with Michael Rapoport, we utilize the fact that
ball quotients can be viewed as moduli spaces of abelian varieties to define arithmetic analogues of the cycles $Z_x$.
We conjecture that the classes in arithmetic Chow groups defined by such cycles are, again, the Fourier
coefficients of certain modular forms. I will discuss some evidence for such a conjecture, especially in the
case of arithmetic $0$-cycles.
Sept. 3, 2010
Peter Shalen :
3 p.m. in SEO 636
Abstract
Hyperbolic geometry is the non-Euclidean geometry discovered by
Lobachevsky, Bolyai and Gauss. In hyperbolic space, the area of a
triangle is determined by the sum of its angles, and more generally
the volume of a configuration is determined by its shape. This
phenomenon, which contrasts with the Euclidean situation, persists in
hyperbolic manifolds, which are metric spaces locally isometric to
hyperbolic space, and are natural objects from the point of view of
differential geometry, complex analysis and number theory. The volume
of a hyperbolic manifold is determined by its topological type.
For hyperbolic manifolds of dimension at least 3, even more is true:
it follows from the Mostow rigidity theorem that when n is at least 3,
a compact hyperbolic n-manifold is determined up to isometry by its
topological type. In dimension 3 the situation is better still: over
the last 30 years, through the efforts of Thurston, Perelman and
others, a complete unification between the topology of 3-manifolds and
the geometry of hyperbolic 3-manifolds has been achieved.
In view of this it is not surprising that topological methods are
powerful in the study of hyperbolic 3-manifolds. My talk will be
focused on a couple of recent results of my own in which topological
methods from classical 3-manifold topology are brought to bear on a
geometric question. In the hyperbolic setting these techniques are
seen to be even richer and more powerful than might have been imagined
when the topology of 3-manifolds was a relatively self-contained and
isolated field.
In the course of describing this work I will try to hint at what a
rich field of research 3-dimensional hyperbolic geometry has become,
involving interactions among geometry, topology, algebra, number
theory and analysis.
Sept. 10, 2010
Sergei Chmutov :
3 p.m. in SEO 636
Abstract
Legendrian knots are knots in a 3-manifold with an additional structure, contact structure. This is a non-integrable field of tangent 2-planes. A Legendrian knot tangents the 2-planes at each point. A transversal knot is transverse to the 2-planes at each point. The presence of the contact structure refines the topological equivalence of knots. For each topological type of knots there are a Legendrian and a transversal representatives. There are simple invariants of Legendian and traversal equivalences withing the same topological type. More delicate invariants come from modern homology theories, contact homology, Khovanov homology and Floer homology. The talk with be a survey of the area of Legendrian and transversal knots. I will explain the front diagrams of Legendrian knots, the braid diagrams of transversal knots, formulate the versions of the Reidemeister moves for these knots, define the simplest invariants, and survey the modern homological invariants.
Sept. 17, 2010
Yuri Tschinkel :
3 p.m. in SEO 636
Abstract
I will explain how to estimate volumes of height balls in analytic varieties over local fields and in adelic points of algebraic varieties over number fields, relating the Mellin transforms of height functions to Igusa integrals and to global geometric invariants of the underlying variety. In the adelic setting, this involves the construction of general Tamagawa measures.
(joint work with A. Chambert-Loir)
Sept. 24, 2010
Alex Wilkie :
3 p.m. in SEO 636
Abstract
I shall introduce the theory of o-minimal structures as a
possible rigorous framework for Grothendieck's idea of "tame topology".
I shall present some of the main finiteness theorems for certain
("definable") subsets of Euclidean space and conclude with an application
concerning the counting of rational points on transcendental analytic
sets.
Oct. 8, 2010
Shi Jin :
3 p.m. in SEO 636
Oct. 15, 2010
Adrian Ioana :
3 p.m. in SEO 636
Abstract
From every countable group G or measure preserving action of G on a probability space X, one can construct a von Neumann algebra. A central theme in the theory of von Neumann algebras is understading how much of the group or group action is ``remembered'' by its von Neumann algebra. In this talk, I will survey recent results which provide the first classes of groups and group actions that can be completely recovered from their von Neumann algebras.
Oct. 21, 2010
Simon Thomas :
3 p.m. in SEO 636
Abstract
Gromov's geometric group theory seeks to classify
finitely generated groups in terms of the ``large scale''
geometry of their Cayley graphs. At first glance, this program
appears to be even more difficult that that of classifying finitely
generated groups up to isomorphism. In this talk, after
introducing some of the basic notions of geometric group
theory and descriptive set theory, I will
consider the question of whether this is indeed the case.
Oct. 22, 2010
S. R. Srinivasa Varadhan :
3 p.m. in SEO 636
Abstract
In this joint work with Sourav Chatterjee, we consider a random graph with $n$ vertices, with probability $p$ for any given edge to be present. We consider the case when $p$ is fixed and $n$ gets large. Asymptotically the expected number of edges is $~{1\over 2}n^2$ and the expected number of triangles is $~{1\over 6}n^3$. We look at the large deviation probabilities for these numbers as well as other subgraph counts. This allows us to say, for example, what a graph with more than the normal number of triangles will look like. The model exhibits an interesting phase transition.
Oct. 29, 2010
Chiu-Chu Melissa Liu :
3 p.m. in SEO 636
Abstract
We extend the familiar assignment in toric geometry which
associates polytopes to line bundles on toric varieties
to an equivalence of categories between equivariant coherent
sheaves on a toric n-fold and constructible sheaves on R^n.
The equivalence -- the coherent-constructible correspondence
-- can be viewed as a categorification of Morelli's description
of equivariant K-theory of toric varieties in terms of a polytope
algebra. This talk is based on joint work with Bohan Fang, David
Treumann and Eric Zaslow.
Nov. 5, 2010
Benson Farb :
3 p.m. in SEO 636
Abstract
Tom Church and I were doing some cohomology computations in topology when we discovered what looked like a pattern. After some struggle, we found a language in which to describe this pattern, and called it "representation stability". We started to look around and soon realized that this phenomenon occurs all over the place, from classical representation theory (Littlewood--Richardson and Murnaghan rules, stability of Schur functors), to cohomology of groups (pure braid, Torelli and congruence groups), to Lie algebras, to the study of flag and Schubert varieties, to algebraic combinatorics (the $(n+1)^{n-1}$ conjecture).
The goal of this talk will be to explain representation stability through an example. I will also explain how Church, Jordan Ellenberg and I are applying this theory in order to compute various combinatorial statistics in number theory. I will try to make this talk accessible to first-year graduate students.
Nov. 12, 2010
Catherine Sulem :
3 p.m. in SEO 636
Abstract
We consider a situation in which a fluid is composed of two essentially immiscible layers
separated by a sharp interface such as a thermocline or a pycnocline of differential salinity.
Internal waves of various types are commonly generated in the world's oceans, and large
amplitude, long wavelength nonlinear waves can be produced in the interface and propagate
over large distances. In some physically realistic instances, the visible signature of internal
waves on the surface of the ocean is a band of roughness which propagates at the same
velocity as the internal wave. Several of the earliest observations are the most striking,
consisting of long brightly shining strips of many kilometers in extent, visible through the
effect of the reflection at an oblique angle of the setting sun, and photographed from the
Space Shuttle. I will discuss the asymptotic analysis of the coupling between the interface and the free surface of a two layers fluid in a scaling regime chosen to capture these observations, in
which the internal mode is treated as a long wavelength nonlinear internal wave, while the
surface mode is smaller and taken in a modulational regime.
This talk is based on joint work with Walter Craig and Philippe Guyenne.
This is the opening lecture for the Midwest PDE Seminar at UIC (November 13-14, 2010).
Nov. 19, 2010
Jacob Rasmussen :
3 p.m. in SEO 636
Abstract
Given a smooth manifold $M$ and a primitive homology class $x \in H_2(M)$, what is the minimal genus of a smoothly embedded surface representing $x$? I'll discuss what we know about this question, and describe applications to topology in dimension three (understanding knots with lens spaces surgeries) and four (constructing exotic $\mathbb R^4$'s).
Dec. 3, 2010
Jacob Fox :
3 p.m. in SEO 636
Abstract
Let $H$ be a fixed graph with $h$ vertices. The graph removal lemma
states that every graph on $n$ vertices with $o(n^h)$ copies of $H$ can be made
$H$-free by removing $o(n^2)$ edges. The graph removal lemma has many
applications in graph theory, additive combinatorics, discrete geometry,
and theoretical computer science.
We give a new proof which avoids Szemeredi's regularity lemma and gives a
better bound. This answers questions of Alon and Gowers.
Jan. 21, 2011
Nigel Higson :
3 p.m. in SEO 636
Abstract
Let K be a closed subgroup of a Lie group G. The contraction of G to K is a Lie group, usually more elementary in structure than G itself, that approximates G to first order near K. The terminology is due to the mathematical physicists, who examined the group of Galilean transformations as a contraction of the group of Lorentz transformations. My focus will be on a related but different class of examples, the prototype of which is the group of isometric motions of Euclidean space, viewed as a contraction of the group of isometric motions of hyperbolic space. It is natural to expect some sort of limiting relation between representations of the contraction and representations of G. But in the 1970s George Mackey carried out a few calculations pointing to an interesting rigidity phenomenon: as the contraction group is deformed back to G, the representation theory remains in some sense unchanged. In particular the irreducible representations of the contraction group parametrize the irreducible representations of G. I shall formulate a reasonably precise conjecture (partly inspired by subsequent developments in C*-algebra theory and noncommutative geometry) and describe the evidence in support of it, which is by now substantial. However a conceptual explanation for Mackey's rigidity phenomenon remains elusive.
Jan. 28, 2011
Nathan Dunfield :
3 p.m. in SEO 636
Abstract
As with many areas of topology and geometry, a starting point in the study of 3-manifolds is to try to understand codimension one objects in them, namely embedded surfaces. A particularly useful class of surfaces are the "incompressible" ones which are topologically essential; a 3-manifold containing such a surface is called a Haken manifold. There are many 3-manifolds which are not Haken, but if we ask about immersed, rather than embedded, surfaces the situation becomes much more mysterious. A closely related question is this: Suppose M is a 3-manifold with infinite fundamental group, does M have a finite cover which is Haken? The Virtual Haken Conjecture posits that the answer to this question is yes.
This talk will survey some recent results in this area, focusing on my work with (variously) William Thurston, Dylan Thurston, Frank Calegari, and Dinakar Ramakrishnan. From the point of view of Thurston's Geometrization Conjecture, this is really a question about hyperbolic 3-manifolds, that is, lattices in PSL(2, C). This opens the door to a rich array of tools that might seem quite surprising in light of the purely topological description of the problem above. Indeed, some unlikely-sounding terms that I will probably mention in my talk are "the Classification of Finite Simple Groups" and "the Langlands Conjecture", as well as such topological oddities as "random 3-manifolds"!
Feb. 4, 2011
Dhruv Mubayi :
3 p.m. in SEO 636
Abstract
The problem of determining the independence number
of hypergraphs has tight connections to questions in discrete geometry,
coding theory, number theory, theoretical computer science and combinatorics. One of the most
famous examples is the result of Komlos-Pintz-Szemeredi (1982) on
the independence number of 3-uniform hypergraphs which made
important progress on the decades old Heilbronn problem. I will begin by explaining this problem and some of these connections. I will then describe a recent result which gives a sharp upper bound on the chromatic number of simple
k-uniform hypergraphs. The talk will be accessible to a general mathematical audience, including graduate students.
Feb. 11, 2011
Daniel Groves :
3 p.m. in SEO 636
Abstract
There are many reasons to study sets of the form Hom(H,G), where H and
G are groups. We'll focus on the situation where G is a fixed group
of interest, and H is allowed to be arbitrary (say finitely
generated). I'll discussion the motivation for studying these sets,
and the focus on the case where G = Mod(S), the mapping class group of
a surface S. In this case, one motivation comes from understanding
surface bundles. This will be explained, and then a structure theory
for Hom(H,Mod(S)) will be hinted at.
Feb. 18, 2011
Davesh Maulik :
3 p.m. in SEO 636
Abstract
Given a family of complex algebraic varieties, there are different ways of making rigorous the notion of a generic member of the family. In most situations, it is easy to see that these approaches lead to the same behavior. However, for small algebraically closed fields, (e.g. the field of algebraic numbers) this distinction is more subtle, due to the countability of the field. In this talk, we give examples of geometric questions where these issues arise and some different techniques for handling them, when possible. As a special case, we will discuss the behavior of Picard groups in families, following ideas of Y. André, and joint work with B. Poonen.
Feb. 25, 2011
Steven Sperber :
3 p.m. in SEO 636
Abstract
Given a family of varieties defined over a finite field of characteristic p or a family of exponential sums we consider for each fiber the set of reciprocal zeros and poles for the zeta function or L-function. We construct new L-functions from this collection of algebraic numbers by taking a product (over the fibers) of factors that have the symmetric powers (or tensor powers, or exterior powers) of the numbers (or a subset of them) in our set as reciprocal zeros. In a joint work with Haessig, we study these L-functions for a general family of non-degenerate toric exponential sums and obtain bounds for the degree of the L-function and a rudimentary estimate for the p-divisibility of the reciprocal zeros and poles of these L-functions.
March 4, 2011
Alexander Kiselev :
3 p.m. in SEO 636
Abstract
Active scalars appear in many problems of fluid dynamics. The most common examples of active scalar equations
include 2D Euler equation describing two-dimensional flow of ideal fluid, surface quasi-geostrophic equation
which appears in atmospheric science, and Burgers equation which is perhaps the simplest active scalar.
Many questions about regularity and properties of solutions of these equations remain open. I will talk about
the recently introduced idea of nonlocal maximum principle and progress it led to. I will also discuss some
questions that remain open, and how these questions relate to other problems of interest in mathematical
fluid mechanics.
March 11, 2011
Volodymyr Nekrashevych :
3 p.m. in SEO 636
Abstract
We will define a notion of a hyperbolic groupoid (or a pseudogroup)
which includes as partial cases actions of Gromov hyperbolic groups on
their boundaries, pseudogroups generated by hyperbolic complex
rational functions, one-sided shifts of finite type, and groupoids
naturally associated with Anosov diffeomorphisms. For every hyperbolic
groupoid G a naturally defined dual groupoid G' acts on the boundary
of the Cayley graphs of G. The dual groupoid G' is also hyperbolic and
G'' is equivalent to G. Different examples of pairs of mutually dual
groupoids, and some applications of the duality theorem will be
described.
March 30, 2011
Various speakers :
3 p.m. in SEO 636
April 1, 2011
Calixto Calderon :
3 p.m. in SEO 636
Abstract
Series of Jacobi Orthogonal Polynomials is always a hot topic in Analysis.
In this lecture I will discuss some not very well known facts, namely, the
two versions for the Poisson Kernel of the Abel Summability of Jacobi
Series. The versions I refer to are the Watson and Bailey kernels. The use
of the Watson form provides good localization results. A theory of
Mukenhoupt weights will be discussed in the context of localization
results. The Bailey's form is nonnegative, and therefore it provides
simple proofs of the norm approximations of the Abel sums. A comment on
early results will be provided.
April 8, 2011
Jacques Verstraete :
3 p.m. in SEO 636
Abstract
If V is a vector space over a finite field F, then a set X of vectors of V is "k-wise independent" if no set of at most k vectors from X is linearly dependent. In this talk I will discuss the problem of determining the maximum size f(n,k,r) of a k-wise independent set of vectors of Hamming weight r in a vector space of dimension n over F. Three simple examples in coding theory, derandomization and the satisfiability problem will be given to motivate this problem.
Following this, a sketch of the proof due to the author and Naor determining the asymptotic value of log f(n,k,r)
as n and k tend to infinity. In words, this guarantees a constant size linear dependency in any set of vectors of weight at most r of size roughly larger than the square root of the Hamming ball of radius r. Using this result, we answer a question of Erdos, Sos and Sarkozy in combinatorial number theory. We close with three open questions.
In part joint work with Assaf Naor
April 15, 2011
Jean-Claude Saut :
3 p.m. in SEO 636
Abstract
The Navier-Stokes equation is both a fundamental equation in Fluid
Mechanics and a rich source of challenging mathematical problems. One
of them is the global existence of "regular" solutions in the
three-dimensional case.
We will make a brief review of some outstanding open problems and
then focus on some recent results linking the asymptotics of the
Navier-Stokes solutions in a particular regime to the normal form
theory of Poincaré and Dulac.
April 22, 2011
Melvin Leok :
3 p.m. in SEO 636
Abstract
The geometric approach to mechanics serves as the theoretical underpinning of innovative control methodologies in geometric control theory. These techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion, and are the basis for understanding the ability of a falling cat to always land on its feet, even when released in an inverted orientation.
We will discuss the application of geometric structure-preserving numerical schemes to the optimal control of mechanical systems. In particular, we consider Lie group variational integrators, which are based on a discretization of Hamilton's principle that preserves the Lie group structure of the configuration space. In contrast to traditional Lie group integrators, issues of equivariance and order-of-accuracy are independent of the choice of retraction in the variational formulation. The importance of simultaneously preserving the symplectic and Lie group properties is also demonstrated.
Recent extensions to homogeneous spaces yield intrinsic methods for Hamiltonian flows on the sphere, and have potential applications to the simulation of geometrically exact rods, structures and mechanisms. Extensions to Hamiltonian PDEs and uncertainty propagation on Lie groups using noncommutative harmonic analysis techniques will also be discussed.
We will place recent work in the context of progress towards a coherent theory of computational geometric mechanics and computational geometric control theory, which is concerned with developing a self-consistent discrete theory of differential geometry, mechanics, and control.
This research is partially supported by NSF grants DMS-0726263, DMS-1001521, and DMS-1010687 (CAREER Award).
April 29, 2011
Vladimir Sverak :
4 p.m. in SEO 636
Abstract
General solutions of the Navier-Stokes equations are complicated not well-understood. In this talk we will dectribe some examples of behavior of the solutions, and we will discuss results about existence and non-existence of some special types of solutions.
Sept. 9, 2011
Michael Aschbacher :
3 p.m. in SEO 636
Abstract
Fusion systems were introduced by Luis Puig as a tool in modular representation theory.
Later homotopy theorists became interested in the work; in particular Broto, Levi,
and Oliver defined the notion of a p-local finite group (consisting of a saturated fusion
system and a linking system), and Levi and Oliver proved the existence of the exotic
Solomon-Benson 2-local finite groups, which in some ways are analogous to sporadic
finite simple groups.
Most of the talk will consist of an introduction to fusion systems, with technicalities
suppressed whenever possible. Near the end of the talk, I'll speculate about the possibility
of simplifying the proof of the classification of the finite simple groups using fusion
systems.
Oct. 21, 2011
Todor Tsankov :
3 p.m. in SEO 636
Abstract
Abstract harmonic analysis and representation theory have traditionally been developed on locally compact groups, where the Haar measure and the regular representation provide the basic tools. However, symmetry groups of infinite combinatorial objects or infinite-dimensional spaces are usually not locally compact, and yet their linear representations occur naturally in various contexts. In this talk, I will describe examples of such groups and survey some old and new classification results for their representations.
Oct. 28, 2011
Jeffrey Rauch :
3 p.m. in SEO 636
Abstract
Parametric resonance occurs when a family time dependent dynamics is explosive while each fixed time problem is conservative or dissipative. We describe historical examples including Turing's instability. And, the recent solution of the Cooper-Strauss conjecture that this phenomenon occurs for the linear wave equation with positive compactly supported smooth potentials periodic in time. Research with F. Colombini and V. Petkov.
Nov. 11, 2011
François Dahmani :
3 p.m. in SEO 636
Abstract
In the continuation of Hilbert's Xth problem on diophantine
equations, Dehn proposed in 1912 three fundamental problems in
(finitely generated) group theory; the word problem, the conjugacy
problem and the isomorphism problem. The questions are to detect when
two given elements of a given group are the same, are conjugate, and
when two given groups are isomorphic. Although, as it was discovered
in the 50's, these problems are algorithmically undecidable in
general, in presence of geometry of negative curvature, one can
usually solve the word problem and the conjugacy problem somewhat
easily. The problems of equations, and of isomorphism are harder in
many respects, but their study has highlighted some deep structures,
and helped introduce powerful tools.
Nov. 18, 2011
Christof Sparber :
3 p.m. in SEO 636
Abstract
We review several recent results on a class of mono-kinetic phase space measures arising from the Bohmian interpretation of quantum mechanics. In particular we shall be concerned with the corresponding classical limit and draw several comparisons to the well known theory of Wigner measures.
Jan. 27, 2012
Jozef Przytycki :
3 p.m. in SEO 636
Abstract
While homology theory of associative structures, such as groups and rings, has been extensively studied in the past beginning with the work of Hopf, Eilenberg,
and Hochschild, homology of non-associative distributive structures, such as quandles, were neglected until recently. Distributive structures have been studied for a long time.
In 1880, C.S. Peirce emphasized the importance of (right) self-distributivity in algebraic structures. However, homology for these universal algebras was introduced only
15-20 years ago by Fenn, Rourke, and Sanderson. We develop this theory in the historical context and propose a general framework to study homology of distributive structures.
We illustrate the theory by computing some examples of 1-term, 2-term, and 3-term homology, and then discussing 4-term homology for Boolean algebras and distributive lattices.
We outline potential relations to Khovanov homology, via the Yang-Baxter operator.
Feb. 3, 2012
Yiming Bao :
3 p.m. in SEO 636
Abstract
The Viral Genome Project (http://www.ncbi.nlm.nih.gov/genomes/GenomesHome.cgi?taxid=10239) at the National Center for Biotechnology Information (NCBI) is a collection of completed genome sequences of viruses with the aim to provide molecular standards for viral genomic research. The project has produced over 4,000 records for more than 2,780 different species. For each virus species, one complete genome is selected as the reference sequence and the rest are marked as its neighbors. The reference sequences are manually curated to correct/update annotations of the original sequence records. Analytical tools provide researchers with the ability to analyze and compare viral genomes and proteomes in different scale in a fast and convenient manner. These tools include Global Alignment of Genome Neighbors, Pairwise Sequence Comparison (PASC), gMAP and Protein Clusters for viral genomes. The Viral Genomes Project is a collaborative effort between NCBI staff and many dedicated scientists worldwide.
In addition, resources for important viruses such as influenza viruses were created. The NCBI Influenza Virus Resource (http://www.ncbi.nlm.nih.gov/genomes/FLU/) provides a curated database that contains nucleotide, protein and coding region sequences of influenza viruses extracted from GenBank. It also has sequence analysis tools that are integrated with the database, such as multiple sequence alignment, clustering of protein sequences, and influenza genome annotation.
Pairwise sequence comparison (PASC): a web tool for virus classification
Pairwise sequence comparison is a sequence-based virus classification method. It calculates the pairwise identities of virus sequences within a virus family and displays their distributions, and can help determine demarcations at different taxonomic levels such as strain, species, genus and subfamily levels. Although this method cannot be used as the single criterion for virus classification in some cases, it is a quantitative method and has many advantages over conventional virus classification methods. It has been applied to polioviruses, coronaviruses, potyviruses, geminiviruses, flexiviruses, papillomaviruses and poxviruses. There is an increasing interest to use this method for other virus families/groups. The PAirwise Sequence Comparison (PASC) tool was created at National Center for Biotechnology Information (NCBI). The tool's database was established with distributions of identity for complete genomes/segments of about 50 virus families/groups. Data in the system are updated periodically to reflect changes in virus taxonomy and additions of new virus sequences to the public database. The web interface of the tool (http://www.ncbi.nlm.nih.gov/sutils/pasc/) makes it easy to navigate and perform analyses. Up to 25 new viral genome sequences can be tested simultaneously with this system within a few minutes to suggest the taxonomic position of the virus isolates in a specific family. This system eliminates potential discrepancies in the results caused by different algorithms and/or different data used by researchers. The NCBI's PASC analysis result for the family Polyomaviridae has been adopted by the ICTV study group as one of the demarcation criteria for new species in the family.
Feb. 10, 2012
Anna Mazzucato :
3 p.m. in SEO 636
Abstract
I will discuss recent results concerning the analysis of incompressible fluid flows at very low viscosity in the presence of walls. Dating back to the pioneering work of Prandtl, such flows are modeled as inviscid away from the walls, but near the walls viscosity effects cannot be neglected, which give rise to a boundary layer where the flow is potentially violent and vorticity is created.
Understanding the viscous boundary layer is of fundamental importance in many physical phenomena, for instance it creates lift around flying objects. A rigorous analysis of boundary layers is still lacking except in special cases. I will present a few classes of flows where such analysis is possible.
Feb. 17, 2012
Peter Keevash :
3 p.m. in SEO 636
Abstract
Matching theory is a large field with many directions of research, both in practical algorithms and combinatorial theory. In this talk I will aim to show some of the breadth of the subject, and some recent advances on matching theory for hypergraphs (joint work with Richard Mycroft). Informally speaking, we show that the obstructions to perfect matchings are geometric, and are of two distinct types: `space barriers' from convex geometry, and `divisibility barriers' from arithmetic lattice-based constructions. We apply our theory to the solution of two open problems on hypergraph packings: the minimum degree threshold for packing tetrahedra in 3-graphs, and Fischer's conjecture on a multipartite form of the Hajnal-Szemeredi Theorem.
March 2, 2012
Catharina Stroppel :
3 p.m. in SEO 636
Abstract
In this talk I will explain the main idea, motivation and a few
consequences of the concept of categorification. All this will be
illustrated by an example coming from knot theory and low dimensional
topology. It turns out that often invariants of knots or links and
3-manifols are given by certain rational numbers which would like to be
categorified, or interpreted as an Euler characteristics of some
complex. The first steps of a possible answer will be explained.
The concept behind this is very general and uses ideas appearing in
representation theory and algebraic geometry.
March 16, 2012
Tom Bohman :
3 p.m. in SEO 636
Abstract
A central theme of probabilistic combinatorics is the study of concentration of measure phenomena in probability spaces populated by interesting combinatorial objects. Dynamic concentration is a variation on this theme in which random variables remain concentrated around their expected trajectories as an underlying process evolves. In this talk we present recent results on dynamic concentration of some processes that have applications in extremal combinatorics.
April 20, 2012
David Damanik :
3 p.m. in SEO 636
Abstract
We will discuss Barry Simon's subshift conjecture, which states that the (OPUC or Schrödinger) spectrum associated with a minimal aperiodic subshift has zero Lebesgue
measure. The first part of the talk will address the history of this problem, from the origins in Kohmoto-Kadanoff-Tang's 1983 paper introducing the Fibonacci Hamiltonian,
through Kotani's 1989 observation that the conjecture follows from the absence of non-uniform hyperbolicity, to proofs of certain kinds of uniformity in certain settings by
Furman in 1997 and Damanik-Lenz in 2006. In the second part we will present recent joint work with Artur Avila and Zhenghe Zhang that produces a counterexample to the
conjecture.
April 27, 2012
William Stein :
3 p.m. in Lecture Center F6
Sept. 14, 2012
Izzet Coskun :
3 p.m. in SEO 636
Abstract
The Hilbert scheme of $n$-points $X^{[n]}$ is a
`compactification' of the set of unordered $n$-tuples of distinct
points on a projective manifold $X$. When the dimension of $X$ is
two, $X^{[n]}$ is a smooth, projective manifold with many remarkable
properties. Consequently, it plays a central role in many areas of
mathematics ranging from representation theory to algebraic geometry
and from algebraic combinatorics to symplectic geometry. For
example, $X^{[n]}$ provide examples of holomorphic symplectic
manifolds when $X$ is a holomorphic, symplectic surface. The homology
of $X^{[n]}$ has the structure of an irreducible representation of
the Heisenberg superalgebra. Similarly, $X^{[n]}$ play a central role
in resolutions of canonical surface singularities. In this talk, I
will give a brief introduction to the Hilbert scheme of points and
its many amazing properties.
Sept. 21, 2012
Amie Wilkinson :
3 p.m. in SEO 636
Abstract
In the early 1930's, the Ergodic theorems of von Neumann
and Birkhoff put Boltzmann's Ergodic Hypothesis in mathematical terms,
and the natural question was born: is ergodicity the "general case"
among conservative dynamical systems? Oxtoby and Ulam tackled this
question early on and showed that the answer to this question is "yes"
for continuous dynamical systems. The work of Kolmogorov, Arnol'd and
Moser beginning in the 1950's showed that the answer to this question
is "no" for $C^{\infty}$ dynamical systems. I will discuss recent work
with Artur Avila and Sylvain Crovisier that addresses what happens for
$C^1$ dynamical systems.
Oct. 5, 2012
Alex Furman :
3 p.m. in SEO 636
Abstract
In this talk I will discuss some classical results and recent developments
in the study of such dynamical systems as automorphisms of the torus
$T^d=R^d/Z^d$ or translations on $SL_d(R)/SL_d(Z)$.
These very concrete systems have interesting dynamics, that have
applications in Number Theory (Margulis's proof of Mahler's conjecture,
the progress by Einsiedler-Katok-Lindenstrauss towards Littlewood's
conjecture, etc.)
In the talk we shall describe some of the known dynamical phenomena that
occur in these systems.
Oct. 19, 2012
Richard Hain :
3 p.m. in SEO 636
Abstract
In this talk I will discuss the beginnings of a theory of
characteristic classes of rational points of smooth projective curves.
This theory is analogous to the theory of characteristic classes of
vector bundles in which grassmanians are replaced by moduli spaces of
curves. I will concentrate on the case where $C$ is defined over the
function field of another curve $T$. In this case, the curve corresponds
to a family $X\to T$ of smooth projective curves over $T$. Rational
points of $C$ correspond to sections of the family $X\to T$. Such
families are classified by maps from $T$ into the moduli space of curves
and rational points of $C$ correspond to lifts of this map to the moduli
space of 1-pointed curves. Mapping class groups are groups of isotopy
classes of diffeomorphisms of a compact oriented surface. They enter
the story as the cohomology of these moduli spaces is the cohomology
of mapping class groups.
Oct. 25, 2012
Dr. Zalman Usiskin :
5 p.m. in SEO 636
Abstract
The words "understand", "understanding", and their plurals appear over 250 times in the Common Core State Standards for Mathematics. With so many appearances, it is not surprising that these words are used in a variety of ways. A framework will be offered to analyze and to assist teachers and students to deal with this variety.
Oct. 26, 2012
Irina Nenciu :
3 p.m. in SEO 636
Abstract
We consider a Schrödinger operator on a bounded domain in R^n,
and search for optimal growth criteria for the potential close to the
boundary of the domain insuring essential self-adjointness of the
associated operator. We find an abstract integral criterion for the
potential, from which we prove that one can add optimal logarithmic
type corrections to the classical criteria. As a consequence of our method,
we study the question of confinement of spinless and spin 1/2 quantum
particles on the unit disk in R, and achieve magnetic confinement solely
by means of the growth of the magnetic field.
Nov. 2, 2012
Benny Sudakov :
3 p.m. in SEO 636
Abstract
Extremal Combinatorics is one of the central branches of discrete
mathematics which deals with the problem of estimating the maximum possible
size of a combinatorial structure which satisfies certain restrictions.
Often, such problems have also applications to other areas including
Theoretical Computer Science, Additive Number Theory and Information Theory.
In this talk we will illustrate this fact by several closely related
examples focusing on a recent work with Alon and Moitra.
Nov. 9, 2012
Menachem Magidor :
3 p.m. in SEO 636
Abstract
The Continuum Problem is whether there is a set of reals whose cardinality is strictly between the cardinality of the integers and the reals.
This was the first problem on Hilbert's famous list and it turned out to be undecidable by the usual axiom systems for Set Theory.
The results of Goedel and Cohen tell us that the axioms give very little information about the relative size of the set of integers and the set of reals.
Goedel's conjecture that strong axioms of infinity will settle the problem turned out to be false. Is this the end of the story?
In this talk we shall survey some of current approaches of trying to give a meaningful answer to the problem, in spite of its independence.
Two directions of research we shall concentrate on will be forcing axioms and the theory of universally Baire sets of reals.
Nov. 16, 2012
Burt Totaro :
3 p.m. in SEO 636
Abstract
The Hodge conjecture predicts which rational homology classes
on a smooth complex projective variety can be represented by linear
combinations of complex analytic subvarieties. In other words,
it is about the difference between topology and complex analysis
(or algebraic geometry). The integral Hodge conjecture,
the analogous conjecture for integral homology classes,
is false in general. We discuss negative results
and some new positive results on the integral Hodge conjecture
for 3-folds.
Feb. 15, 2013
Jared Wunsch :
3 p.m. in SEO 636
Abstract
The principle of geometric optics postulates that "wavefronts" of solutions to the wave equation move along classical particle trajectories (i.e., geodesics). This principle has its limitations, though. For instance, we cannot localize a solution to the wave equation along a single closed geodesic forever: it eventually leaks away (essentially owing to the uncertainty principle). I will discuss some recent progress in quantifying the degree to which waves can be concentrated in regions of phase space invariant under a particle's dynamics.
Feb. 22, 2013
Mattias Jonsson :
3 p.m. in SEO 636
Abstract
The common zero set of a collection of complex polynomials defines an analytic object, in nice cases a complex manifold. Berkovich spaces appear when trying to do the same thing for polynomials with coefficients in a non-Archimedean field, such as p-adic numbers or Laurent series.
I will explain how Berkovich spaces can sometimes be used to study problems over the complex numbers and discuss some non-Archimedean analogues of results in complex geometry.
March 1, 2013
Gregory Eyink :
3 p.m. in SEO 636
Abstract
We review the notion of "spontaneous stochasticity", which
arose from
the work of Richardson (1926) on turbulent 2-particle dispersion and
which corresponds
to a breakdown in uniqueness of solutions to ODE's with vector fields
only Hoelder
continuous in space. This phenomenon has been rigorously established
in probabilistic
models of Brownian flows (Kraichnan model) and evidence of the same
phenomenon
is observed in empirical data for Navier-Stokes turbulence at high
Reynolds numbers.
For the scalar advection-diffusion equation (Lie transported 0-forms)
``spontaneous
stochasticity" implies a robustly unique class of weak solutions which
dissipate energy.
We discuss how "spontaneous stochasticity" influences the turbulent
kinematic magnetic
dynamo (Lie-transported 1-forms), both by analytical results for
Brownian flows and by
Lagrangian numerical studies for Navier-Stokes turbulence. There is a
close analogy
between the ideal magnetic induction equation for 1-forms and the
incompressible Euler
equation for the velocity. This analogy suggests several natural
conjectures for fluid
circulations in the weak solutions of the Euler equations that
describe fluid turbulence,
as conjectured by Onsager. These conjectures are supported also by
recent work of
Constantin & Iyer (2008), which implies equivalence of the
incompressible Navier-Stokes
equation to a stochastic Kelvin Theorem.
March 8, 2013
Jack Cowan :
3 p.m. in SEO 636
Abstract
We have recently found a way to describe large-scale neural activity in terms of non-equilibrium statistical mechanics. This allows us to calculate the effects of fluctuations and correlations on neural activity. Major results of this formulation include a role for critical branching, and the demonstration that there exists a non-equilibrium phase transition in neocortical activity, which is in the same universality class as directed percolation. Here we show how the population dynamics of interacting excitatory and inhibitory neural populations can be described in similar terms, and how such a theory can be used to explain the origins and properties of random bursts of synchronous activity (avalanches), population oscillations (quasi-cycles), synchronous oscillations (limit-cycles) and fluctuation-driven spatial patterns (quasi-patterns). If time permits, we will also describe recent work on a way to incorporate spike-timing dependent plasticity into a model for self-organized criticality.
March 15, 2013
Ming-Jun Lai :
3 p.m. in SEO 636
Abstract
I will start with a motivation how to recover a low-rank matrix from
a small number of its linear measurements, e.g., a subset of its
entries. As such problems share many common features with the recent
study of recovering sparse vectors in compressed sensing, I shall
give a quick review with some most updated research results on sparse vector
recovery and matrix completion.
Then I will explain an unconstrained $L^q$ minimization approach and an
iteratively reweighted algorithm for recovering sparse vectors as well
as for recovering low-rank matrices. A convergence
analysis of these iterative algorithms will be given.
Finally, I shall present some numerical results for recovering images
from their random sampling entries without and with noises.
March 22, 2013
Tom Bridgeland :
3 p.m. in Lecture Center C4 (please note special location)
Abstract
A celebrated theorem of Ringel gives a neat formula-free
description of positive parts of quantized enveloping algebras in terms of
categories of quiver representations. The crucial ingredient is the notion
of the Hall algebra of an abelian category. I will explain this
construction, give a precise statement of Ringel's result, and then
describe a generalization which gives a similar formula-free description
for the full quantum group.
April 5, 2013
Ilijas Farah :
3 p.m. in SEO 636
Abstract
G. Mackey introduced the notion of smooth equivalence relation in the context of unitary group representations.
Over the last two decades Mackey's notion was extended to an abstract framework for analyzing classification problems in mathematics.
Methods developed by Hjorth, Kechris and others enable one to precisely define when one classification problem is more difficult than another.
Classification problems are compared using the relation of Borel-reducibility.
I will outline some of the recent results, putting a particular emphasis on applications to Elliott's program for classification of nuclear C*-algebras.
This is a joint work with a number of coauthors.
April 12, 2013
Dhruv Mubayi :
3 p.m. in SEO 636
Abstract
Finite extremal set theory is concerned with the following general problem: Suppose we have a collection F of subsets of
an n-element set and we have some restriction on the possible intersection sizes of pairs of sets in F.
What is the maximum number of subsets that F can contain? Surprisingly, solutions to various special cases of this problem have deep implications in many other areas,
including coding theory, geometry, and computer science. A particular famous example is due to Frankl and Rodl, who solved a 250-dollar problem of Erdos by proving
that if n is a multiple of 4 and n/4 is excluded as an intersection size, then |F| < (1.99)^n. We extend this result by showing that if some additional (rather mild) restrictions
are placed on the possible intersection sizes, then |F|< (1.63)^n. This is joint work with Vojtech Rodl.
The talk will be accessible to a general mathematical audience, including graduate students.
April 19, 2013
Ryan Martin :
3 p.m. in SEO 636
Abstract
In a statistical inference problem, the goal is to describe uncertainties about the truthfulness of various hypotheses after seeing data. There are now a variety of ways to do this, but none are fully satisfactory. In this talk, I will describe a brand new approach, what we call "inferential models" (IMs). The key idea is the introduction of an auxiliary variable connected with the observable data and parameter of interest. We employ random sets to predict the auxiliary variable, producing a belief and plausibility function pair on the parameter space that can be used to summarize uncertainty. I will show that a surprisingly simple "nestedness" condition on the random sets is both sufficient and, in a certain sense, necessary for the resulting IM to be "good." (This is joint work with Chuanhai Liu at Purdue.)
April 26, 2013
Lek-Heng Lim :
3 p.m. in SEO 636
Abstract
This talk is intended for those who, like the speaker, have at
some point wondered whether there is a theory of three- or higher-
dimensional matrices that parallels matrix theory. We will discuss how
notions like rank, norm, determinant, eigen and singular values may be
generalized to hypermatrices. We will see that, far from being artificial
constructs, these notions have appeared naturally in a wide range of
applications: chemistry (fluorescence spectroscopy, density matrix
renormalization group), computer science (matrix multiplication
complexity, quantum computing), optimization (self-concordance,
higher-order optimality conditions), statistics (higher-order moments and
cumulants, minimum rank matrix completion), physics (quark states,
Yang-Baxter equations), and signal processing (antenna array processing,
CDMA radio communication).
May 3, 2013
Akshay Venkatesh :
3 p.m. in Lecture Center F4 (please note special location)
Abstract
I will discuss some models of what a "random abelian group"
is, and some conjectures (the Cohen-Lenstra heuristics of the title)
about how they show up in number theory. I'll then discuss
the function field setting and a proof of these heuristics,
with Ellenberg and Westerland. The proof is an example of a link between analytic number theory and certain classes of results in algebraic topology ("homological stability").
Sept. 13, 2013
Mimi Dai :
3 p.m. in SEO 636
Abstract
We consider the Ericksen-Leslie model for nematic liquid crystal (LCD) systems. Regularity and uniqueness of solutions to the density dependent LCD system were established. In two dimension, global regular solutions exist with general data; in three dimension, global regular solutions exist with the assumption of small initial data and short time regular solutions exist for large data. In addition, with more smoothness assumption on initial data, we obtain the uniqueness both for dimension 2 and 3 cases. Furthermore, in the case of constant density, the long time behavior of regular solutions was studied and optimal decay (time) rate was obtained for the solutions in all of the Sobolev spaces $H^m$ with $m \geq 0$.
Sept. 20, 2013
Lev Reyzin :
3 p.m. in SEO 636
Abstract
I will describe the planted clique problem, a famous problem at the intersection of combinatorics and computer science. I will discuss progress on this problem, as well as recent hardness results we've been able to prove. I will also talk about its relationship to the problem of solving linear equations over GF(2), via what are known as "statistical queries".
Oct. 4, 2013
Anand Pillay :
3 p.m. in SEO 636
Abstract
I discuss a theory of ``definable topological dynamics" namely of definable actions of a definable group on compact spaces, a special case of which is the classical abstract topological dynamics of a discrete group. The notions of topological dynamics provide new invariants for groups definable in first order theories. Conversely the model theoretic perspective provides on the face of it some new invariants for discrete groups.
I will also discuss examples, such as semisimple real and p-adic Lie groups, considered as (real, p-adic) semialgebraic groups.
Oct. 11, 2013
David Nualart :
3 p.m. in SEO 636
Abstract
The fractional Brownian motion is a centered self-similar Gaussian process with stationary increments, which depends on a parameter $H$ in $(0,1)$ called the Hurst index. We will first describe some basic properties of the fractional Brownian motion such as long-range dependence and finite p-variation. The applications of the fractional Brownian to model data coming from engineering, finance and other areas, require the construction of a suitable stochastic calculus, similar to the classical Ito calculus. In this talk we review some recent results on the stochastic calculus with respect to the fractional Brownian motion with emphasis on the construction of stochastic integrals using different types of Riemann sums approximations. We will present central limit results for critical values of the Hurst parameter where the approximation diverges, and we will discuss numerical approximation schemes for stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H>1/2$.
Oct. 25, 2013
John Stufken :
3 p.m. in SEO 636
Abstract
A certain combinatorial arrangement, now known as an orthogonal array, was introduced for use in statistics in the 1940's. The primary reason for its introduction was to guide the selection of level combinations in a fractional factorial experiment, and this remains an important application. Various criteria based on statistical properties have been introduced over the years to distinguish between different orthogonal arrays of the same size, and some authors have attempted to enumerate all non-isomorphic arrays of small sizes. Orthogonal arrays also possess interesting relationships to several other combinatorial arrangements, including error-correcting codes and Hadamard matrices. In this talk, we will present a brief overview of orthogonal arrays, including their existence, construction, and relationships to other arrangements.
Nov. 1, 2013
Tsachik Gelander :
3 p.m. in SEO 636
Abstract
We study the asymptotic behavior of the Betti numbers of higher rank locally symmetric manifolds as their volumes tend to infinity, and prove a uniform version of the Lueck Approximation Theorem, which is much stronger than the linear upper bounds proved by Gromov. The basic idea is to adapt the theory of local convergence, originally introduced for sequences of graphs of bounded degree by Benjamimi and Schramm, to sequences of Riemannian manifolds. Using rigidity theory we are able to show that when the volume tends to infinity, the manifolds locally converge to the universal cover in a sufficiently strong manner that allows one to derive the convergence of the normalized Betti numbers. Similarly, and more generally, we show that the normalized multiplicity of any unitary representation converges to its Plancherel measure.
For congruence covers of a fixed manifold, we also obtain sharper estimates in terms of the rate of convergence.
Joint work with M. Abert, N. Bergeron, I. Biringer, N. Nikolov, J. Raimbault and I. Samet.
Nov. 8, 2013
Ridgway Scott :
3 p.m. in SEO 636
Abstract
We discuss both continuum and atomic scale models
of solvation and the hydrophobic effect. We describe their
application to modeling of protein-ligand interactions. A
key result is the development of a nonlocal model that can
be computed in only twice the time required for the standard
model, despite the fact that formally the nonlocal model is
a six-dimensional PDE.
Nov. 15, 2013
Various Panelists :
3 p.m. in SEO 636
Abstract
Ever wonder what the AWM is, and why we need it? Come join the UIC AWM chapter for an oral history discussion about mathematics and women. This event is open to all, in particular, visitors from other universities and departments are encouraged to attend.
<br>
The panelists are Bhama Srinivasan (UIC), Jeanne LaDuke (DePaul), Anne Leggett (Loyola), Alexandra Bellow (Northwestern). One more panelist, a contemporary woman mathematician, is to be confirmed. The panel will be moderated by Steve Hurder (UIC).
<br>
Jeanne LaDuke is a historian of mathematics and the author (with Judy Green) of "Pioneering Women in American Mathematics: pre-1940 Ph.D's". Anne Leggett is the long-time editor of the AWM Newsletter and the editor (with Bettye Anne Case) of "Complexities: Women In Mathematics". Alexandra Bellow is a distinguished mathematician who was a Professor at Northwestern University from 1967 until 1996.
Nov. 22, 2013
Stephen Stigler :
3 p.m. in SEO 636
Abstract
This year marks the 300th anniversary of the publication of
Jacob Bernoulli1s book, Ars Conjectandi, and the 250th anniversary of the
reading of Thomas Bayes1s famous work to the Royal Society of London.
Bernoulli1s book is widely know as the source for the Law of Large
Numbers, while Bayes1s article is regarded as the original introduction of
Bayesian inference. It might be supposed that after so many years of
intensive study there is nothing new to be said about either. This
supposition is incorrect, and the surprising results of recent research
will be presented that may help deepen our understanding of both.
Jan. 31, 2014
Ben Weinkove :
3 p.m. in SEO 636
Abstract
The Ricci flow has been a powerful tool in the study of
three-dimensional manifolds. I will discuss the behavior of this flow
on an important class of four-dimensional manifolds: the Kahler
surfaces. In addition, I will discuss a flow which generalizes the
Kahler-Ricci flow called the Chern-Ricci flow. This is a geometric
flow on complex manifolds, recently introduced by M. Gill. I will
describe some recent results and conjectures in the case of complex
surfaces.
Feb. 7, 2014
Isaac Goldbring :
3 p.m. in SEO 636
Abstract
The notion of existentially closed structure is a model-theoretic generalization of the notion of algebraically closed field. A common goal in model theory is to try and understand the existentially closed objects of some axiomatizable category. In this talk, I will explain an interesting connection between the search for an existentially closed C* algebra with a certain important property (namely exactness) and a conjecture of Kirchberg concerning the ultrapower of the Cuntz algebra $\mathcal{O}_2$. All relevant model theoretic and C* algebraic notions will be defined.
Feb. 21, 2014
Brendan Hassett :
3 p.m. in LC C6
Abstract
Holomorphic symplectic manifolds are higher-dimensional analogues of K3
surfaces. In the 1990's Huybrechts established when these are projective
varieties, but a precise description of possible embeddings emerged only in
the last couple years. We will survey recent developments including new
theorems drawing on results on stability conditions for derived categories.
(This is joint work with Bayer and Tschinkel.)
March 7, 2014
Michael Hopkins :
3 p.m. in LC D5
Abstract
http://www.math.uic.edu/seminars/poster?seminar_id=3027
March 14, 2014
Ramin Takloo-Bighash :
3 p.m. in SEO 636
Abstract
In this talk I will explain various conjectures about the distribution of rational points on algebraic varieties,
and illustrate them using concrete examples. I will also explain several classes of examples where these conjectures have been verified. This talk is based on joint works with Alex Gorodnick, Joseph Shalika, and Yuri Tschinkel.
I will try to make the talk accessible to first year graduate students.
March 21, 2014
Rakesh Vohra :
3 p.m. in SEO 636
Abstract
I will survey some results about the accuracy of prediction when the predictor has no prior knowledge about the process that s/he must forecast. No prior knowledge means just that; no moments, distributions, periods etc. For example, suppose one is asked to predict successive outcomes of an infinite sequence of 0's and 1's. Accuracy will be measured by the fraction of correct guesses. With no information beyond this, how well can one guarantee to do? Predicting the actual outcome is demanding and in many cases inappropriate; think for example of the case when the sequence is generated by a stochastic process. In this case it is more natural to ask for a probability forecast. How should one measure the error of a probability forecast? Given this measure, are there forecasting algorithms that guarantee a small error no matter what process generates the sequence?
April 4, 2014
Matthew Foreman :
3 p.m. in SEO 636
Abstract
This talk describes joint work with with B. Weiss showing that it is impossible to classify diffeomorphisms of the torus up to conjugacy by measure preserving transformations. A new functor is presented that maps odometer based transformations and their joinings to transformations based on the circle and their joinings.
The talk will point out connections with the problem of classifying diffeomorphisms up to conjugation by homeomorphisms.
April 11, 2014
Jeff Brock :
3 p.m. in SEO 636
Abstract
The classification of hyperbolic 3-manifolds with finitely generated fundamental group involves
sorting a lot of pretty ugly looking bugs. Geometrically, this classification reduces to
Thurston's <i>Ending Lamination Conjecture</i>; our proof (with Canary and Minsky) produced an
array of tools for understanding how a topological specification of a 3-manifold relates to
its "shape" or geometry. One such specification, the <i>Heegaard splitting</i>, seems ripe for
exploration along these lines, yet a complete picture is still under development. I'll
review the state of things, and discuss some key examples.
April 18, 2014
Douglas Arnold :
3 p.m. in SEO 636
Abstract
This talk will discuss a substantial interplay of algebraic
topology with numerical analysis which has developed over the
last decade. During this period, de Rham cohomology and the Hodge
theory of Riemannian manifolds have come to play a crucial role
in the development and understanding of computational algorithms
for the solution of problems in partial differential equations.
Hodge theory is at the foundation of the well-posedness of many
important problems in partial differential equations, but only
recently has it been understood how the stable numerical solution
of PDE problems often depends on capturing the correct topological
structures at the discrete level. This may be accomplished by
constructing subcomplexes of the de Rham complex which consist
of finite element differential forms. These latter have become
a key technology in scientific computation, but first appeared
in algebraic topology in the works of Whitney and Sullivan half a
century ago. Interactions of topology and numerical analysis are
not just applications of the former to the latter, but also occur
in the reverse direction. Recently, the theory of superconvergence
developed for finite element methods was used to settle a question
concerning the combinatorial codifferential posed by Dodziuk and
Patodi in 1976.
April 25, 2014
Mircea Mustata :
3 p.m. in SEO 636
Abstract
Toric varieties are algebraic varieties endowed with a "nice"
action of an algebraic torus. A remarkable feature is that their geometry
can be fully described in terms of combinatorics of fans and polytopes.
After explaining what these objects are and some classical facts about
their cohomology, I will discuss some results concerning the topology of
the fibers of toric maps and a combinatorial invariant that comes out of
these considerations. This is based on joint work in progress with Marc de
Cataldo and Luca Migliorini.
May 2, 2014
Showu Zhang :
3 p.m. in LC C4
Abstract
A thousand years old problem is to determine which
positive integers are congruent numbers,
i,e, those numbers which could be the areas of right angled
triangles with sides of rational lengths.
This problem has some beautiful connections with elliptic curves and
L-functions.
In fact by the Birch and Swinnerton-Dyer conjecture, all n= 5, 6, 7
mod 8 should congruent numbers,
and most of n=1, 2, 3 mod 8 should not not congruent numbers.
In this lecture, I will explain these connections and then some recent progress
based on the Waldspurger formula and the Gross--Zagier formula.
Aug. 29, 2014
Douglas Arnold :
3 p.m. in SEO 636
Abstract
This talk will discuss a substantial interplay of algebraic
topology with numerical analysis which has developed over the
last decade. During this period, de Rham cohomology and the Hodge
theory of Riemannian manifolds have come to play a crucial role
in the development and understanding of computational algorithms
for the solution of problems in partial differential equations.
Hodge theory is at the foundation of the well-posedness of many
important problems in partial differential equations, but only
recently has it been understood how the stable numerical solution
of PDE problems often depends on capturing the correct topological
structures at the discrete level. This may be accomplished by
constructing subcomplexes of the de Rham complex which consist
of finite element differential forms. These latter have become
a key technology in scientific computation, but first appeared
in algebraic topology in the works of Whitney and Sullivan half a
century ago. Interactions of topology and numerical analysis are
not just applications of the former to the latter, but also occur
in the reverse direction. Recently, the theory of superconvergence
developed for finite element methods was used to settle a question
concerning the combinatorial codifferential posed by Dodziuk and
Patodi in 1976.
Sept. 26, 2014
Rodrigo Bañuelos :
3 p.m. in SEO 636
Abstract
In October 1910 Hendrik Antoon Lorentz, 1902 Nobel Prize in Physics, delivered a series of six lectures (the Paul Wolfskehl Lectures) to the faculty of the University of Göttingen titled "old and new problems in physics." During the fourth lecture, with David Hilbert and his student Hermann Weyl present in the audience, he conjectured that the number of eigenvalues for the Laplacian for a region $D$ in three space not exceeding the positive number $\lambda$ is proportional to the volume of $D$ times $\lambda^{3/2}$, when $\lambda$ is large. (The problem had been raised a month earlier by Arnold Sommerfeld at a lecture in Könisberg.) Hilbert predicted that the conjecture would not be proved in his lifetime. He was wrong by several years. The conjecture was proved by Weyl in 1912.
Weyl's celebrated theorem, commonly referred to as <i> Weyl's Law</i>, has been extended and refined in many directions with connections to many areas of mathematics and physics. In this talk we first give an overview of some of the classical results in the field and discuss the elegant connections to Brownian motion first explored by Mark Kac in the 50's and 60's. We will then discuss problems that arise when the Brownian motion, which "goes" with the classical Laplacian, is replaced by other Lévy processes. Such processes share many important properties with Brownian motion. We will look at a class of interesting examples that have been widely studied recently, the rotationally invariant stable processes that "go" with fractional powers of the Laplacian.
Oct. 3, 2014
Danny Calegari :
3 p.m. in SEO 636
Abstract
In 1985, Barnsley and Harrington defined a "Mandelbrot Set" M for pairs of similarities - this is the set of complex numbers z with norm less than 1 for which the limit set of the semigroup generated by the similarities x → zx and x → z(x-1)+1 is connected. Equivalently, M is the closure of the set of roots of polynomials with coefficients in {-1,0,1}. Barnsley and Harrington already noted the (numerically apparent) existence of infinitely many small "holes" in M, and conjectured that these holes were genuine. These holes are very interesting, since they are "exotic" components of the space of (2 generator) Schottky semigroups. The existence of at least one hole was rigorously confirmed by Bandt in 2002, but his methods were not strong enough to show the existence of infinitely many holes; one difficulty with his approach was that he was not able to understand the interior points of M, and on the basis of numerical evidence he conjectured that the interior points are dense away from the real axis. We introduce the technique of *traps* to construct and certify interior points of M, and use them to prove Bandt's Conjecture. Furthermore, our techniques let us certify the existence of infinitely many holes in M. This is joint work with Sarah Koch and Alden Walker.
Oct. 24, 2014
Jianguo Sun :
3 p.m. in SEO 636
Abstract
Survival analysis is one of major and important fields
in statistics and this is especially true from the point of biological and medical
research. In addition, we also see the increasing of its applications
in many other fields including demography, economics finance,
political science, psychology and sociology. In this talk, we will first
give some basic and simple introduction of survival analysis and
then discuss several current research topics in the field related
to the analysis of interval-censored survival data, a special and
common type of survival data.
Oct. 31, 2014
De Witt Sumners :
3 p.m. in SEO 636
Abstract
This talk will survey some of the results on properties of random
knots in 3-space and in confined volumes, with applications to enzyme action on
duplex DNA and the structure and dynamics of duplex DNA confined to viral
capsids. This talk is intended for a general mathematical audience.
Nov. 7, 2014
Wilfrid Gangbo :
3 p.m. in SEO 636
Abstract
We construct a small time strong solution to a nonlocal Hamilton-Jacobi equation introduced by Lions, the so-called master equation, originating from the theory of Mean Field Games. We discover a link between metric viscosity solutions to local Hamilton-Jacobi equations studied independently by Ambrosio-Feng and G-Swiech, and the master equation. As a consequence we recover the existence of solutions to the First Order Mean Field Games equations, first proved by Lions. We make a more rigorous connection between the master equation and the Mean Field Games equations. (This talk is based on a joint work with A. Swiech).
Nov. 14, 2014
Jun Liu :
3 p.m. in SEO 636
Abstract
I will discuss a few recent results from my group aiming to the detection
of non-linear dependence between two random variables. Our approach is
based on an optimal slicing (discretization) of one or both variables to
optimize a score function derived from a likelihood-ratio test formulation.
Our approaches are compared with some well-known methods such as Distance
Correlation, Pearson Correlation, Maximal Information Criterion, etc., on
many simulated examples, and found superior for highly nonlinear and
non-smooth relationships between the two variables. We will also show how
these methods are applied to bioinformatics problems such as gene-set
enrichment analysis, transcription regulation analysis, etc.
Feb. 27, 2015
Birgit Richter :
3 p.m. in SEO 636
March 13, 2015
Steven Zelditch :
3 p.m. in SEO 636
Abstract
Quantum mechanics solved the problem of how an electron
can be moving and stationary at the same time, by replacing the
classical motion of the electron by the wave function. The wave
function satisfies Schrödinger's eigenvalue problem, formally like
the equation for vibrational modes of a drum. But how can the
stationary state be 'visualized' and related to the motion of the
classical particles as the Planck constant tends to zero?
<br>
My talk will review this background and then discuss new results (jointly
with J.J. Jung, John Toth, and/or Chris Sogge)
on the size and shapes of stationary states. In particular, I will discuss
$L^p$ norms of eigenfunctions and the distribution of the nodal (zero) sets
of eigenfunctions. Nodal sets are somewhat similar to real algebraic
varieties of degree equal to the square root of the eigenvalue and most
problems and conjectures regarding nodal sets are similar to ones in
real algebraic geometry.
April 10, 2015
Joe Harris :
3 p.m. in SEO 636
Abstract
An elementary theorem says that we can always find a polynomial $f(x)$ of degree $d$ or less having specified values at $d+1$ given points $x$. When we try to state (let alone prove) an analogue for polynomials in several variables, however, we run into immediate difficulties. In this talk, I’ll try to show that the difficulties lie in the geometry of the points, and suggest at least a conjectural answer to the problem.
April 17, 2015
Carlos Kenig :
3 p.m. in SEO 636
Abstract
We will describe some recent works on the soliton resolution
conjecture, for nonlinear wave equations. The soliton resolution
conjecture, in this setting states that a general solution, asymptotically
in time, decomposes as a finite sum of modulated solitons and radiation.
In our recent works (with Duyckaerts and Merle, and with Lawrie, Liu and
Schlag) we have proven this for the energy critical wave equation in the
radial case, in 3d, and for equivariant exterior wave maps, also in 3d.
April 24, 2015
Steven Leth :
3 p.m. in SEO 636
Abstract
In this talk I will outline some applications of nonstandard methods to the
study of compact, connected subsets of the plane. Nonstandard models allow
for many complicated limiting properties of a set to be "actualized" in the nonstandard version of the set. This
can make objects such as "pseudo-arcs" and
other hereditarily indecomposable continua more intuitive to work with. Of
particular interest are possible applications to sub-questions of the
<i>plane fixed point problem</i>, which asks if every compact, connected
subset of the plane that does not separate the plane has the fixed point
property.
May 1, 2015
Alireza Salehi-Golsefidi :
3 p.m. in TBA
Aug. 28, 2015
M. Ram Murty :
3 p.m. in SEO 636
Abstract
We give a survey lecture on the development of
sieve theory and then discuss recent developments initiated
by Yitang Zhang, Maynard and Tao that approach the
twin prime problem. At the end of the talk,
we will highlight the ``higher rank Selberg sieve'' that
emerges from these developments and discuss recent joint
work with Akshaa Vatwani. The talk will be accessible
to a wide audience.
Oct. 2, 2015
Dale Cutkosky :
3 p.m. in SEO 636
Abstract
We discuss some good properties of analytic mappings, including flatness, regularity and being locally monomial.
All of these properties can be obtained locally after performing suitable sequences of local blow ups.
Key notions are that of a star and the stellar vault, which were introduced by Hironaka.
Oct. 9, 2015
Jason Hartline :
3 p.m. in SEO 636
Abstract
Computer systems have become the primary mediator of social and
economic interactions. A defining aspect of such systems is that the
participants have preferences over system outcomes and will manipulate
their behavior to obtain outcomes they prefer. Such manipulation
interferes with data-driven methods for designing and testing system
improvements. A standard approach to resolve this interference is to
infer preferences from behavioral data and employ the inferred
preferences to evaluate novel system designs.
In this talk I will describe a method for estimating and comparing the
performance of novel systems directly from behavioral data from the
original system. This approach skips the step of estimating
preferences and is more accurate. Estimation accuracy can be further
improved by augmenting the original system; its accuracy then compares
favorably with ideal controlled experiments, a.k.a., A/B testing,
which are often infeasible. A motivating example will be the
paradigmatic problem of designing an auction for the sale of
advertisements on an Internet search engine.
Oct. 16, 2015
David Marker :
3 p.m. in Lecture Center C4
Abstract
In the 90s model theorists introduced the notion of
o-minimal geometry to find more general settings where results from
real algebraic geometry could be extended. In recent years these
ideas have found applications in Pila's work on the Andre-Oort
Conjecture. I will survey some of these ideas and Pila and Zannier's
application to give a new proof of the Manin-Mumford Conjecture.
Oct. 23, 2015
Marianna Csornyei :
3 p.m. in SEO 636
Abstract
One of the classical theorems of Lebesgue tells us that Lipschitz
functions on the real line are differentiable almost everywhere. We study
possible generalisations of this theorem and some interesting geometric
corollaries.
Oct. 30, 2015
Ryan Martin :
3 p.m. in SEO 636
Abstract
Statistical methodology has made extraordinary advances in recent years, but the foundations of statistics still are not yet fully developed. In fact, basic questions such as "what is statistical inference?" remain unanswered. In this talk, I will present a definition of statistical inference, introduce a key validity criterion, and relate these ideas to what has been called the "most important unsolved problem in statistics". After giving some background to put the problem in perspective, I will introduce the new <i>inferential model</i> (IM) framework and argue that it solves that important unsolved problem. Some examples will be presented to demonstrate the potential of IMs, and I will conclude with some important open problems, interesting (perhaps) to both statisticians and mathematicians.
Nov. 2, 2015
Peter Semrl :
3 p.m. in SEO 636
Abstract
Two matrices are said to be adjacent if their difference is of
rank one. Fundamental theorems of geometry of matrices proved by L.-K.
Hua describe the general form of bijective maps on various spaces of
matrices preserving adjacency in both directions. We will present
several recent improvements of these results and some applications in
mathematical physics.
Nov. 6, 2015
Davar Khoshnevisan :
3 p.m. in SEO 636
Abstract
We present some recent optimal regularity results for parabolic SPDEs driven by space-time white noise.
Our results are connected closely to questions about how sensitive the solution to a parabolic SPDE is to small changes in the initial data.
This is based on joint work with Le Chen and Kunwoo Kim
Nov. 20, 2015
Noga Alon :
3 p.m. in SEO 636
Abstract
The study of Cayley sum-graphs of finite abelian groups is related to the investigation of pseudo-random graphs and to problems in Combinatorial Number Theory, Geometry and Information Theory. I will discuss this topic, describing the motivation
and focusing on several results that illustrate the interplay between Graph Theory, Geometry and Number Theory.
Dec. 4, 2015
David Nicholls :
3 p.m. in SEO 636
Abstract
The interaction of waves (acoustic, electromagnetic, elastic) with layered
media plays an important role in many scientific problems. Among these are
seismic imaging, underwater acoustics, biosensing, and solar cells. The
ability to simulate scattered fields from these structures in a robust and
highly accurate fashion is of fundamental importance. In this talk we will
describe a class of rapid and highly accurate boundary perturbation schemes,
termed ``High--Order Perturbation of Surfaces'' (HOPS) Methods, for
delivering such numerical approximations. Time permitting we will describe
our efforts to not only detect layered media geometries based upon far--field
data, but also design these structures to have optimal scattering properties.
Jan. 22, 2016
Ramin Takloo-Bighash :
3 p.m. in SEO 636
Abstract
The talk will start with some remarks on the role that zeta functions and
Tauberian theorems have played in number theory in the last 180 years starting
essentially with Dirichlet's proof of his Arithmetic Progression Theorem.
The remainder of the talk will be devoted to giving a survey of recent
applications of Tauberian theorems to counting arithmetic objects.
Feb. 5, 2016
Mimi Dai :
3 p.m. in SEO 636
Abstract
An important feature of dissipative systems is the existence of the global
attractor that describes the long-time behavior of all the solutions. When
the spacial domain is not bounded and Poincaré’s inequality is not
valid, the existence of the global attractor is still an open question.
However, when the force is small, one can prove that the global attractor
is a unique fixed point using the Fourier splitting method. We apply this
method to study the long-time behavior of solutions to various fluid
equations including the Navier-Stokes, and certain complex fluid models,
such as the liquid crystal systems, and obtain optimal decay rates for the
solutions.
Feb. 12, 2016
Professor Joel E. Cohen :
3 p.m. in SEO 636
Abstract
"Taylor's law" asserts that, in sets of samples of a nonnegative quantity
(e.g., insect population abundance), the sample variance is approximately
proportional to some power of the sample mean. Taylor's law has been verified
for hundreds of species and in many fields beyond ecology, including physics
and finance. As scientific motivation, I will show some empirical examples of
Taylor's law from my own work. The main focus of my talk is the different
mathematical interpretations of Taylor's law and the great diversity of
theoretical models (from stochastic processes, differential equations,
and number theory, among other areas of mathematics) that lead to Taylor's law.
Feb. 19, 2016
Zhen-Qing Chen :
3 p.m. in SEO 636
Abstract
Boundary theory for one-dimensional diffusions is now well understood. Boundary theory for multi-dimensional diffusions is much richer and remains to be better understood. In this talk, we will be concerned with the construction and characterization of obliquely reflected Brownian motions in all bounded simply connected planar domains, including non-smooth domains,
with general reflection vector field on the boundary.
We show that the family of all obliquely reflected Brownian motions in a given domain can be characterized in two different ways, either by the field of angles of oblique reflection on the boundary or by the stationary distribution and the rate of rotation of the process about a reference point in the domain. We further show that Brownian motion with darning and excursion reflected Brownian motion can be obtained as a limit of obliquely reflected Brownian motions.
Based on joint work with K. Burdzy, D. Marshall and K. Ramanan.
Feb. 26, 2016
Ingrid Daubechies :
3 p.m. in Cardinal Room 329 of Student Center East
Abstract
Mathematics can help Art Historians and Art Conservators in studying and understanding art works,
their manufacture process and their state of conservation. The presentation will review several instances of
such collaborations in the last decade or so, and then focus on one particular example: virtual cradle removal.
Between the 12th to the 17th century, European artists typically painted on wooden boards. To remediate or
prevent structural or insect damage, conservators in the 19th and first half of the 20th century first thinned
the panels to a few mm, and then strengthened the much thinner wood structures by (permanently) attaching
to their backs hardwood lattices called cradles. These cradles are highly visible in X-ray images of the paintings.
X-rays of paintings are a useful tool for art conservators and art historians to study the condition of a painting, as well as the techniques used by the artist and subsequent restorers. The cradling artifacts obstruct a clear ``reading'' of the X-rays by these experts.
These artifacts can be removed, using a variety of mathematical tools, including Bayesian algorithms.
March 4, 2016
1. Laura Schaposnik, 2. Kevin Tucker :
3 p.m. in SEO 636
Abstract
This is a special Open House Colloquium aimed at our regular faculty and students
and at the prospective students considering to join our graduate program.
The Colloquium will consist of two talks above.
L. Schaposnik: Moduli spaces may be thought of as geometric solutions to
geometric classification problems. Along the talk we shall first introduce
these type of problems by working through some toy examples: by considering the moduli space of lines we will understand how lines may be classified, how line bundles can be obtained,
and finally how to define Higgs bundles.
K. Tucher: It's a fact of life -- even if you're only interested smooth
geometric objects, they tend to degenerate to singular (read: non-manifold) ones.
In this talk, I'll try to give an idea what some singular algebraic varieties
really "look like." This will lead us to a discussion of invariants of
singularities and some topics of current research.
March 11, 2016
Gyorgy Turan :
3 p.m. in SEO 636
Abstract
A directed hypergraph has edges of the form a,b --> c, i.e., edges
have a single head, but can have multiple vertices in the tail. This is a
generalization of directed graphs. Directed hypergraphs are closely
related to Horn formulas in logic, closure operators, lattices and
functional dependencies in databases.
The study of these concepts is an area where `graph theory meets reasoning'.
We discuss directed hypergraphs from the point of view of optimization,
extremal and probabilistic combinatorics, and knowledge representation
and reasoning in artificial intelligence.
April 15, 2016
Thomas Schlumprecht :
3 p.m. in SEO 636
Abstract
For a Banach space $X$ we consider $\mathcal L(X)$, the algebra of linear bounded operators on $X$.
A closed subideal of $\mathcal L(X)$, is a subideal which is closed in the operator norm.
For very few Banach spaces $X$ the structure of the closed subideals of $\mathcal L(X)$ is well
understood.
For example it is known for a long time that the only non trivial closed subideals of
$\mathcal L(\ell_p)$ (other than the zero ideal and the entire algebra) is the ideal of compact operators.
In his book ``Operator Ideals'' Albrecht Pietsch asked about the structure of the closed subideals of
$\mathcal L(\ell_p\oplus\ell_q)$, the space of operators on the complemented sum of $\ell_p$ and $\ell_q$,
where $1\leqslant p < q\leqslant \infty$.
In particular he asked if there are infinitely many closed subideals.
This question was recently solved affirmatively for the reflexive range $1 < p < q < \infty$,
in a joint work by the author in collaboration with Andras Zsak.
April 29, 2016
Henri Gillet :
3 p.m. in SEO 636
Abstract
I shall discuss how one might use Milnor K-theory and Nash functions to model the relationship between the Chow groups, the classical singular cohomology, and the de Rham cohmomology of a non-singular algebraic variety over the complex numbers.
Sept. 16, 2016
Charles Doering :
3 p.m. in SEO 636
Abstract
Buoyancy forces result from density variations, often due to temperature variations, in the presence of gravity.
Buoyancy-driven fluid flows shape the weather, ocean and atmosphere dynamics, the climate, and the structure
of the earth and stars. In 1916 Lord Rayleigh published a paper entitled "On Convection Currents in a Horizontal
Layer of Fluid, when the Higher Temperature is on the Under Side" introducing the minimal mathematical model
of buoyancy-driven fluid flow now known as Rayleigh-Bénard convection. For a century this model has served
as a primary paradigm of complex nonlinear dynamics displaying spontaneous symmetry breaking and pattern
formation, chaos and turbulence. Here we describe progress and challenges for the analysis of Rayleigh's
model in the strongly nonlinear regime of turbulent convection.
Sept. 23, 2016
Todd Kemp :
3 p.m. in SEO 636
Abstract
Random matrix theory studies the behavior of the eigenvalues (or singular values)
of random matrices as the dimension grows. Initiated by Wigner in the 1950s,
there is now a rich and well-developed theory of the universal behavior of such
random eigenvalues in models that are natural generalizations of the Gaussian
case.
In this talk, I will discuss a generalization of these kinds of results in a new
direction. A Gaussian random matrix can be thought of as an instance of Brownian
motion on a Lie algebra; this opens the door to studying the eigenvalues (and
singular values) of Brownian motion on Lie groups. I will present recent
progress understanding the asymptotic spectral distribution of Brownian motion
on unitary groups and general linear groups. The tools needed include
probability theory, combinatorics, and representation theory.
Sept. 30, 2016
Carl Mueller :
3 p.m. in SEO 636
Abstract
Hitting questions play a central role in the theory of stochastic processes. For example, we could consider our wealth as a random process and think of “striking it rich” as an example of this random process hitting the set of rich values. Here is a purely mathematical example. It is well known that Brownian motion hits points in one dimension, but not in higher dimensions. For a general Markov process, we can determine whether the process hits a given set in terms of potential theory. There has also been a huge amount of work on the related question of when a process has multiple points.
For stochastic partial differential equations (SPDE), much less is known, but there has been a growing number of papers on the topic in recent years. Potential theory provides an answer in principle. But unfortunately, solu- tions to SPDE are infinite dimensional processes, and the potential theory is intractible. As usual, the critical case is the most difficult.
We will give a brief survey of known results, followed by a discussion of an ongoing project with R. Dalang, Y. Xiao, and S. Tindel which promises to answer questions about hitting points and the existence of multiple points in the critical case.
Oct. 21, 2016
Richard Canary :
3 p.m. in LC F6
Abstract
It is a classical result that the geometry of a closed hyperbolic
surface is completely determined by the lengths of finitely many simple
closed geodesics on the surface. One may reformulate this in algebraic
language, as saying that a discrete, faithful representation of the fundamental
group G of a closed surface S is determined, up to conjugacy, by the spectral radii
of the images of finitely many elements which are represented by simple closed
curves on S.
Hitchin discovered a component of the space of representations of G into PSL(n,R),
which bears many resemblances to the Teichmuller space of all representations of
G into PSL(2,R). We show that Hitchin representations are similarly determined by the
spectral radii of the images of elements represented by simple closed curves.
We obtain a similar result for discrete faithful representations of G into PSL(2,C).
(These results are joint work with Martin Bridgeman and Francois Labourie.)
Nov. 4, 2016
Michael Overton :
3 p.m. in SEO 636
Abstract
Crouzeix's conjecture is among the most intriguing developments in matrix theory in recent years.
Made in 2004 by Michel Crouzeix, it postulates that, for any polynomial p and any matrix A,
||p(A)|| <= 2 max(|p(z)|: z in W(A)), where the norm is the 2-norm and W(A) is the field
of values (numerical range) of A, that is the set of points attained by v*Av for some
vector v of unit length. Remarkably, Crouzeix proved in 2007 that the inequality above
holds if 2 is replaced by 11.08. Furthermore, it is known that the conjecture holds in a
number of special cases, including n=2. We use nonsmooth optimization to investigate
the conjecture numerically by attempting to minimize the “Crouzeix ratio”, defined as the
quotient with numerator the right-hand side and denominator the left-hand side of the
conjectured inequality. We present numerical results that lead to some theorems and
further conjectures, including variational analysis of the Crouzeix ratio at conjectured global minimizers.
All the computations strongly support the truth of Crouzeix’s conjecture.
This is joint work with Anne Greenbaum and Adrian Lewis.
Nov. 11, 2016
Andrew Suk :
3 p.m. in SEO 636
Abstract
The classic 1935 paper of Erdos and Szekeres entitled "A combinatorial problem in geometry" was a starting
point of a very rich discipline within combinatorics: Ramsey theory. In that paper, Erdos and Szekeres studied
the following geometric problem. For every integer n \geq 3, determine the smallest integer ES(n) such that
any set of ES(n) points in the plane in general position contains n members in convex position, that is, n points
that form the vertex set of a convex polygon. Their main result showed that
ES(n) \leq {2n - 4\choose n-2} + 1 = 4^{n -o(n)}. In 1960, they showed that ES(n) \geq 2^{n-2} + 1 and
conjectured this to be optimal. Despite the efforts of many researchers, no improvement in the order of
magnitude has been made
on the upper bound over the last 81 years. In this talk, we will sketch a proof showing that ES(n) =2^{n +o(n)}.
Feb. 10, 2017
Carlos Kenig :
3 p.m. in SEO 636
Abstract
We will give an overview of the recent developments on the
long-time dynamics for large solutions of the energy critical wave
equation, emphasizing the non-radial case.
Feb. 24, 2017
Mike Jolly :
3 p.m. in SEO 636
Abstract
A determining form for a dissipative partial differential equation is an ordinary differential equation in a certain trajectory space where the solutions on the global attractor of the PDE are readily recognized. It is an ODE in the true sense of defining a vector field which is (globally) Lipschitz. We discuss two types of determining forms: one where solutions on the global attractor of the PDE are traveling waves, and another where they are steady states. Each determining form is related to a certain approach to data assimilation, i.e. the injection of a coarse-grain time series into the model in order to recover the matching full solution. Applications have been made to the 2D incompressible Navier-Stokes, damped-driven nonlinear Schrodinger, damped-driven Korteveg-de Vries and surface quasigeostrophic equations.
March 10, 2017
Lev Reyzin, Dima Sinapova :
3 p.m. in SEO 636
Abstract
There will be two 30min talks.
Part I. Lev Reyzin: A Few TCS Problems on Graphs
I will give a few vignettes, each about a theoretical computer science (TCS) problem that my students
and I have been worked on during the last few years.
One common theme will be that these problems concern graphs, thereby being easier to intuitively
understand in a short talk. I will end each vignette with a tempting open problem.
Part II. Dima Sinapova: Truth and infinity.
Famously, in 1963 Cohen invented the breakthrough method of forcing and used it to show that
the continuum hypothesis is independent of ZFC, resolving Hilbert's first problem.
Since then, a long standing project in set theory has been to use forcing for relative consistency
results and to analyze cardinal arithmetic, especially for singular cardinals.
I will go over some background and then discuss current research in singular combinatorics.
March 31, 2017
James Freitag :
3 p.m. in SEO 636
Abstract
Painleve equations are certain order two nonlinear differential equations which were isolated around the beginning of the last century by Painleve, Gambier, and Fuchs for reasons related to classical analytic problems. The equations arise in a variety of applications from physics to Diophantine geometry. In this talk, we will discuss how model theory can be used to resolve some open problems around the transcendence of Painleve equations.
April 7, 2017
Marta Lewicka :
3 p.m. in SEO 636
Abstract
We consider a free boundary problem for a system of PDEs, modeling the growth of a biological tissue. A morphogen, controlling volume growth, is produced by specific cells and then diffused and absorbed throughout the domain. The geometric shape of the growing tissue is determined by the instantaneous minimization of an elastic deformation energy, subject to a constraint on the volumetric growth. For an initial domain with C^{2,\alpha} boundary, our main result establishes the local existence and uniqueness of a classical solution, up to a rigid motion.
This is a joint work with Alberto Bressan.
April 14, 2017
Jacob Bedrossian :
3 p.m. in SEO 636
Abstract
In this talk we will discuss some of the intricacies of Landau damping in the collisionless Vlasov equations or the collisionless limits of Vlasov-Fokker-Planck equations. We will discuss the construction of solutions to the Vlasov-Poisson equations on S x R which are arbitrarily close to homogeneous equilibrium in Sobolev regularity but which display arbitrarily long sequences of nonlinear oscillations known as plasma echoes. In particular, these oscillations show that the collisionless linearization is not valid for long times in Sobolev regularity. Further, we show that the inclusion of weak collisional effects suppress these plasma echoes and make it possible to obtain Sobolev regularity results. We also prove that Debye shielding and dispersive effects can suppress such nonlinear oscillations (joint with Nader Masmoudi and Clement Mouhot). Combined with the existing infinite regularity results of Mouhot and Villani, these results together confirm and refute a variety of conjectures made by both mathematicians and physicists over the years regarding Landau damping near homogeneous equilibrium.
April 21, 2017
Sándor Kovács :
3 p.m. in Douglas Hall 210
Abstract
Rational singularities play an important role in many parts of algebraic geometry. Their most significant property is that their cohomology theory works very much as if they were regular, but the class of rational singularities is much more robust than that of regular points.
Traditionally, the two fundamental pillars of studying rational singularities over the complex numbers have been:
(i) resolutions of singularities, and
(ii) Kodaira-type vanishing theorems.
In positive characteristic, however, resolutions of singularities may not exist and Kodaira-type vanishing theorems generally fail to hold.
In this talk, I will describe a new approach to rational singularities which do not rely on resolutions as well as a vanishing theorem that is general enough to prove a characteristic independent version of Kempf's criterion for rational singularities.
In turn, this result may be used to prove a characteristic independent version of Elkik's theorem which states that most of the singularities of the minimal model program are rational.
Another application is to counting rational points on varieties defined over a finite field. In particular, I will discuss a generalization of Esnault's theorem on rational points of smooth Fano varieties to mildly singular log Fano varieties which also gives a new proof of Esnault's theorem.
April 26, 2017
Susan Friedlander :
3 p.m. in SEO 636
Abstract
We consider the three dimensional magnetohydrodynamics (MHD) equations in
the presence of stochastic forcing as a model for magnetostrophic
turbulence. For scales relevant to the Earth's fluid core this MHD system
is very rich in small parameters. We discuss results concerning the
asymptotics of the stochastically forced PDEs in the limit of vanishing
parameters. In particular we establish that the system sustains ergodic
statistically steady states thus providing a rigorous foundation for
magnetostrophic turbulence.
This is joint work with Juraj Foldes, Nathan Glatt-Holtz and Geordie
Richards.
April 28, 2017
Yuri Tschinkel :
3 p.m. in TBA
Sept. 1, 2017
Martina Bode and Jenny Ross :
3 p.m. in SEO 636
Abstract
TBA
Sept. 15, 2017
Panagiotis Souganidis :
3 p.m. in SEO 636
Abstract
I will present recent developments about first- and second-order PDE on junctions. I will discuss some concrete applications, the well-posedness of the problems, as well as stability of asymptotic limits including fattening of domains. If time permits, I will also present some new results about scalar conservation laws along the same spirit. This is joint work with Pierre-Louis Lions.
Sept. 22, 2017
Walter Craig :
3 p.m. in SEO 636
Abstract
The evolution of vortex filaments in three dimensions is a question of mathematical hydrodynamics which involves the analysis of nonlinear partial differential equations. On the physical side it is relevant to questions of vortex evolution for the Euler equations as well as to the fine structure of vortex cores in a superfluid. On the mathematical side it is a setting of partial differential equations with a compelling analogy to Hamiltonian dynamical systems. In this lecture I will describe a model for the dynamics of near - parallel vortex filaments and their mutual interactions in a three dimensional fluid. The talk will describe a phase space analysis of solutions, including constructions of periodic and quasi-periodic orbits via a version of KAM theory in an infinite dimensional phase space, and a topological principle to count the multiplicity of solutions. This is ongoing joint work with L. Corsi (Georgia Institute of Technology), C. Garcia (UNAM) and C.-R. Yang (McMaster and Shantou University)
Sept. 29, 2017
Gabor Szekelyhidi :
3 p.m. in SEO 636
Abstract
Kahler-Einstein metrics are of fundamental importance in
Kahler geometry, with connections to algebraic geometry, geometric
analysis, string theory amongst other fields. Their study has received
a great deal of attention recently, culminating in the solution of the
Yau-Tian-Donaldson conjecture, characterizing which complex manifolds
admit Kahler-Einstein metrics. I will give an overview of the field,
including some recent developments.
Oct. 6, 2017
Eitan Tadmor :
3 p.m. in SEO 636
Abstract
Edges are noticeable features in images which can be extracted from noisy data using different variational
models. The analysis of such models leads to the question of representing general L^2-data as the
divergence of uniformly bounded vector fields.
We use a multi-scale approach to construct uniformly bounded solutions of div U=f for
general f’s in the critical regularity space L^2(T^2). The study of this equation and related
problems was motivated by recent results of Bourgain & Brezis. The intriguing critical aspect
here is that although the problems are linear, construction of their solution is not. These
constructions are special cases of a rather general framework for solving linear equations in
representations U=\sum_j u_j which we introduced earlier in the context of image processing,
yielding a multi-scale decomposition of "image" U.
Oct. 13, 2017
Cristian E. Gutierrez :
3 p.m. in SEO 636
Abstract
The refractor problem consists in designing a surface separating two homogeneous media with different refractive indices, that refracts radiation emanating from a point source or a set of sources into a target destination with a prescribed distribution of energy. The solution of this problem yields free form lenses refracting monochromatic radiation in a prescribed manner, and it has connections with the areas of optimal mass transport and Monge-Ampere type equations. In this lecture, I will present the background for the problem and describe some results showing the connections with these areas. I will also describe an algorithm to solve the problem numerically.
Oct. 20, 2017
Min Yang :
3 p.m. in SEO 636
Abstract
Extraordinary amounts of data are being produced in many branches of science. Proven statistical methods are no longer applicable with extraordinary large data sets due to computational limitations. A critical step in Big Data analysis is data reduction. In this presentation, I will review some existing approaches in data reduction and introduce a new strategy called information-based optimal subdata selection (IBOSS). Under linear and nonlinear models set up, theoretical results and extensive simulations demonstrate that the IBOSS approach is superior to other approaches in term of parameter estimation and predictive performance. The tradeoff between accuracy and computation cost is also investigated. When models are mis-specified, the performance of different data reduction methods are compared through simulation studies. Some ongoing research work as well as some open questions will also be discussed.
Oct. 27, 2017
Sergei Starchenko :
3 p.m. in SEO 636
Abstract
Let $A$ be a complex abelian variety and $\pi\colon \mathbb{C}^n\to A$
be the covering map.
It follows from a theorem of Ax that for an irreducible subvariety
$X\subseteq \mathbb{C}^n$ the Zariski closure of $\pi(X)$ is a coset
of an algebraic subgroup of $A$.
In this talk we consider \emph{the topological closure} $\pi(X)$ of an
algebraic subvariety $X$ of $\mathbb{C}^n$ and describe it in terms of
finitely many algebraic families of cosets of real subtori.
We also obtain a similar description when $A$ is a real torus and
$X$ is a semi-algebraic set.
Nov. 3, 2017
Jeffrey Rauch :
3 p.m. in SEO 636
Abstract
To compute approximate solutions of
partial differential equations on all of space one usually performs
computations on a bounded computational domain.
Often the domain is chosen rectangular therefore with
trihedral corners in three dimensional space.
Artificial absorbing boundary conditions are imposed.
One needs to analyse dissipative boundary value
problems with trihedral corners. Existence is easy.
Uniqueness is not. Describe recent work with
Laurence HALPERN and open problems.
Nov. 10, 2017
Nessim Sibony :
3 p.m. in SEO 636
Abstract
Consider the polynomial differential equation in $\mathbb C^2$
$$\frac{dz}{dt}=P(z,w),\qquad \frac{dw}{dt}=Q(z,w).
$$
The polynomials $P$ and $Q$ are holomorphic, the time is complex. In order to study the global
behavior of the solutions, it is convenient to consider the extension as a foliation
in the projective plane $\mathbb P^2$.
I will discuss some recent results around the following questions. What are the tools for ergodic theory in this setting?
What is the ergodic theory of such systems? How do the leaves distribute in a generic case?
The system exhibits some surprising rigidity aspects.
Jan. 18, 2018
Will Perkins :
3 p.m. in SEO 636
Abstract
Random instances of computational problems play an important role in computer science as a source of hard instances, a testbed for algorithms, and as practical constructions of error correcting codes. The biggest breakthrough in our understanding of these problems in the last decade has come from statistical physicists who were originally interested in the properties of glasses. Their intricate but non-rigorous "cavity method” gives a series of detailed predictions of the behavior of randomized computational problems.
I will describe a mathematical vindication of the cavity method in a broad class of models. Using a mix of tools, new and old, we make rigorous the calculations the physicists have been doing all along. Some consequences of our results include determining the information theoretic threshold in the disassortative stochastic block model and the condensation threshold in the random graph coloring problem. Based in part on joint work with A. Coja-Oghlan, F. Krzakala, and L. Zdeborova.
Feb. 16, 2018
Joseph M. Landsberg :
3 p.m. in SEO 636
Abstract
Our story begins with a spectacular failure:
The standard algorithm to multiply two nxn matrices uses $n^3$ multiplications. In 1969, while attempting to show that the standard algorithm was optimal, V. Strassen discovered an explicit algorithm to multiply 2x2 matrices using 7 multiplications rather than $8=2^3$. It is a central question to determine just how efficiently one can multiply nxn matrices, both practically and asymptotically.
In this talk, I will present a history of the problem, both of upper and lower complexity bounds I will discuss how geometry, more precisely algebraic geometry and representation theory,
are used. In particular, I will explain how, had someone asked him 100 years ago, the algebraic geometer Terracini could have
predicted Strassen's algorithm. The talk will conclude with the recent use of representation theory to construct algorithms, more precisely, rank decompositions.
For those who can't wait for the talk, a detailed history and the state of the art appears in Landsberg, J. (2017). Geometry and Complexity Theory (Cambridge Studies in Advanced Mathematics 169).
Feb. 23, 2018
Mirela Ciperiani :
3 p.m. in SEO 636
Abstract
Genus one curves, defined over the rationals, need not have rational points. The set of all such curves, whose Jacobian is a fixed elliptic curve E, forms a group, called the Weil-Chatelet group. It has an important subgroup, the Tate-Shafarevich group, formed by those curves which have points over all completions of the rationals.
This talk will address two aspects of the arithmetic of genus one curves: (1) (with J. Stix) the divisibility of the elements of the Tate-Shafarevich group inside the Weil-Chatelet group; (2) (with A. Wiles) the existence of points defined over number fields with solvable Galois group over the rationals on genus one curves that correspond to elements of the Tate-Shafarevich group; we aim to extend this result to the whole Weil-Chatelet group.
March 9, 2018
Michael Loss :
3 p.m. in SEO 636
Abstract
The goal of kinetic theory is to describe the behavior
of a large collection of colliding particles. In 1956
Mark Kac invented a probabilistic model that describes
a spatially homogeneous gas. This linear model helps to understand
some aspects of kinetic theory, such as molecular chaos which
is closely connected to the non-linear Boltzmann equation
and approach to equilibrium. In this talk I will give a leisurely
introduction to these issues and will also present some newer results.
March 16, 2018
Avrim Blum :
3 p.m. in SEO 636
Abstract
In this talk I will discuss the problem of trying to learn the requirements and preferences of economic agents by observing the outcomes of an allocation mechanism whose rules you also don’t initially know. As an example, consider observing web pages where the agents are advertisers and the winners are those whose ads show up on the given page. We know these ads are placed based on bids and other constraints given to some auction mechanism, but we do not get to see these bids and constraints. What we would like to do is from repeated observations of this type to learn what the requirements and preferences of the agents are. Or consider observing the input-output behavior of some scheduling service, where the input consists of a set of agents requesting service, and the output tells us which actually received service and which did not. In this case, we assume the agents who did not receive service were not served due to overlap of their resource needs with higher-priority requests. From such input-output behavior, we would like to learn the underlying structure. Our goal will be from observing a series of such interactions to try to learn both the needs and preferences of the agents and perhaps also the rules of the allocation mechanism.
This talk is based on work joint with Yishay Mansour and Jamie Morgenstern, as well as work joint with Michael Liang.
March 23, 2018
Joe Harris :
3 p.m. in Lecture Center A1
Abstract
The Brill-Noether theorem establishes a fundamental link between the classical notion of a curve in projective space, given as the zero locus of polynomials, and the (relatively) modern notion of an abstract curve. Specifically, it tells us when and how a given general abstract curve can be embedded in $\mathbb{P}^r$.
But that's just the opening line of the story: having embedded our abstract curve in projective space, we can ask about the geometry and algebra of the image. In particular, we ask what sort of polynomial equations define the image -- what their degrees are, and how many of them there are. <i>The Maximal Rank Conjecture</i>, recently proved by Eric Larson, gives the answer to this question. In this talk, I'll describe the ideas leading up to this theorem, give an overview of the proof, and discuss the questions that follow.
March 30, 2018
Spring Vacation :
3 p.m. in SEO 636
April 6, 2018
Andre Neves :
3 p.m. in SEO 636
Abstract
Minimal surfaces are ubiquitous in geometry and applied science but their existence theory is rather mysterious. For instance, Yau in 1982 conjectured that any 3-manifold admits infinitely many closed minimal surfaces but the best one knows is the existence of at least three.
After a brief historical account, I will talk about my ongoing work with Marques and the progress we made on this question jointly with Irie and Song: we showed that for generic metrics, minimal hypersurfaces are dense and equidistributed. In particular, this settles Yau’s conjecture for generic metrics.
April 13, 2018
Harm Derksen :
3 p.m. in SEO 636
Abstract
If G is a group acting on a vector space V by linear transformations,
then the invariant polynomial functions on V form a ring. In this talk
we will discuss upper bounds for the degrees of generators of this
invariant ring. An example of particular interest is the action of the
group SL_n x SL_n on the space of m-tuples of n x n matrices by
simultaneous left-right multiplication. In this case, Visu Makam and
the speaker recently proved that invariants of degree at most mn^4
generate the invariant ring. We will explore an interesting connection
between this result and the notion of noncommutative rank.
April 20, 2018
Steve Shkoller :
3 p.m. in SEO 636
Abstract
The motion of compressible fluids is difficult to simulate numerically, particular in multiple space dimensions, due to the presence of wave patterns with sharp discontinuities, such as shock waves. In this lecture, I will describe how some ideas from PDE (partial differential equations) can be used to develop accurate numerical methods which allow for highly singular phenomenon, such as shock waves colliding with walls and bouncing back, in a very stable manner. The talk will be fairly self-contained so no prior knowledge of fluids or numerics is necessary, and I will mostly use short videos of our numerical results to demonstrate the schemes.
April 27, 2018
Adrian Diaconu :
3 p.m. in SEO 636
Abstract
L-functions --- vast generalizations of the Riemann zeta-function --- are fundamental objects of study in number theory. In the 1980's the idea emerged that it could be useful to tie
together a family of related L-functions in one variable to create a double, or multiple, Dirichlet series, which could be used to study the average behavior of the original family of L-functions.
The local structure of these multiple Dirichlet series shows a rich connection to the theory of automorphic forms, and representation theory. On the automorphic side, Whittaker functions
on p-adic groups and their covers are the fundamental objects. Whittaker functions and their relatives are expressible in terms of combinatorial structures on the associated L-group, its flag variety, or Schubert varieties. In the combinatorial theory, crystal graphs, Demazure characters, the Schubert calculus and Kazhdan-Lusztig theory all enter.
In this talk, I will focus on the most important case, namely the multiple Dirichlet series associated to moments of L-functions. I will discuss the connection (established recently in joint work with Vicentiu Pasol) between the local parts of these series and the compactifications of certain
moduli spaces of curves, and how this information can be combined with the (conjectural in general) analytic continuation of the multiple Dirichlet series to obtain precise asymptotics for moments, for example, of the classical family of quadratic Dirichlet L-functions.
This talk is designed for a general mathematical audience.
May 4, 2018
Ramin Takloo-Bighash :
3 p.m. in SEO 636
Sept. 7, 2018
Helena J. Nussenzveig Lopes :
3 p.m. in 636 SEO
Abstract
The vanishing viscosity problem consists of understanding the limit, or limits, of solutions of the Navier–Stokes equations, with viscosity $\nu$, as $\nu$ tends to zero. The Navier–Stokes equations are a model for real-world fluids and the parameter $\nu$ represents the ratio of friction, or resistance to shear, and inertia. Ultimately, the relevant question is whether a real-world fluid with very small viscosity can be approximated by an ideal fluid, which has no viscosity. In this talk we will be primarily concerned with the classical open problem of the vanishing viscosity limit of fluid flows in domains with boundary. We will explore the difficulty of this problem and present some known results. We conclude with a discussion of criteria for the vanishing viscosity limit to be a solution of the ideal fluid equations
Sept. 14, 2018
Anush Tserunyan :
3 p.m. in 636 SEO
Abstract
A flourishing subject in modern descriptive set theory is the study of countable Borel equivalence relations on Polish spaces. Each such equivalence relation can always be generated in two ways: as the orbit equivalence relation of a Borel action of a countable group, and as the connectedness relation of a locally countable Borel graph. These strong connections between equivalence relations, group actions, and graphs create an extremely fruitful interplay between descriptive set theory, ergodic theory, measured group theory, percolation theory, and descriptive graph combinatorics. As an example, I will discuss how descriptive set theoretic thinking combined with combinatorial and measure theoretic arguments yields a pointwise ergodic theorem for quasi-probability-measure-preserving locally countable graphs.
Sept. 28, 2018
Holly Krieger :
3 p.m. in 636 SEO
Abstract
I will discuss joint work with Laura DeMarco and Hexi Ye in which we use dynamically-inspired techniques towards a conjecture of Bogomolov-Fu-Tschinkel asserting a uniform bound on the number of common torsion points of distinct elliptic curves. I will explain our strategy, which has general application to proving uniform bounds in unlikely intersections, and how this theorem implies a uniform bound on the number of torsion images in their Jacobians for a family of genus 2 curves. This talk will be accessible to a general mathematical audience.
Oct. 5, 2018
Lihong Zhi :
3 p.m. in 636 SEO
Abstract
The sparse interpolation problem has been studied and widely used in many different areas of science and engineering since the work of Prony (1795). Ankur Moitra in his paper at STOC 2015 has given an in-
depth analysis of how oversampling improves the conditioning of the arising Prony systems for sparse interpolation and signal recovery from numeric data. Moitra assumes that oversampling is done for a number of samples beyond the actual sparsity of the polynomial/signal. We give an algorithm that can be used to compute the sparsity and estimate the minimal number of samples needed in numerical sparse interpolation. Some recent work on computing the symmetric tensor rank by Prony's method will also be introduced.
Oct. 12, 2018
None :
3 p.m. in 636 SEO
Oct. 26, 2018
Alex Wilkie :
3 p.m. in 613 SCE
Abstract
Please note the room for this talk is 613 Student Center East.
Let $K$ be a subfield of the complex field $\mathbb{C}$. By an exponential
polynomial over $K$ we mean a function of the form $P(z_0, \ldots , z_n ,
e^{z_0}, \ldots , e^{z_n})$ where $P$ is a polynomial over $K$. In this
talk I discuss complex space curves (in $\mathbb{C}^{n+1}$) given as the
intersection (assumed everywhere nonsingular with respect to the variables
$z_1, \ldots , z_n$) of the zero sets of $n$ such exponential polynomials.
Let $\Omega$ be such a curve and let $\pi_0 [\Omega]$ be its projection on
to the $z_0$-plane. Then $\pi_0 [\Omega]$ is an open subset of
$\mathbb{C}$ and one expects it to be co-countable. Even this special case
of Zilber's famous conjecture on the complex exponential field (which will
be explained in the talk) is, as far as I know, still unknown. But I shall
present a result that implies that the open set $U \cap \pi_0 [\Omega]$ is
dense (in $U$) and connected for every connected open subset $U$ of
$\mathbb{C}$. This has the model theoretic consequence that the set of
reals does not lie in the $\sigma$-algebra generated by the subsets of
$\mathbb{C}$ defined in the complex exponential field by existential
formulas.
Please note the room for this talk is 613 Student Center East.
Nov. 2, 2018
Jared Weinstein :
3 p.m. in 636 SEO
Abstract
According to Hilbert, the theory of complex multiplication is not only the most beautiful part of mathematics but also of all science. Complex multiplication refers to a lattice in the complex numbers (or an elliptic curve) which admits endomorphisms by a ring larger than the integers. We will begin with Kronecker's "Jugendtraum" -- the use of complex multiplication to solve Hilbert's twelfth problem. This will lead us into a discussion of some fascinating work by Gross and Zagier on the j-invariants of elliptic curves with complex multiplication. We will conclude with some recent work on the modular curve "at infinite level", which is a perfectoid space, and the unexpected role that complex multiplication plays in its geometry.
Nov. 9, 2018
Guillaume Bal :
3 p.m. in 636 SEO
Abstract
Topological insulators(TIs) are materials characterized by topological invariants. One of their remarkable features is the asymmetric transport observed at the interface between materials in different topological phases. Such transport is itself described by a topological invariant, and therefore ``protected" against random perturbations. This immunity makes TIs extremely promising for many engineering applications and actively researched.
In this talk, we present a PDE model for such TIs, introduce a topology based on indices of Fredholm operators, and analyze the influence of random perturbations. We confirm that topology is an obstruction to Anderson localization, a hallmark of wave propagation in strongly heterogeneous media in the topologically trivial case and to some extent quantify what is or is not protected topologically. For instance, a quantized amount of transmission is protected while back-scattering, a practical nuisance, is not.
Nov. 16, 2018
Bridget Tenner :
3 p.m. in 636 SEO
Abstract
We use enumerative and algebraic combinatorics to prove results on a range of topics. These include algebraic structures, tiling phenomena, pattern avoidance, and even applications to political science.
Dec. 7, 2018
Arend Bayer :
3 p.m. in LC D5
Abstract
I will explain how the seemingly highly abstract machinery of derived categories can be used to answer fundamental and concrete questions in algebraic geometry. I will give several examples of this philosophy; the one alluded to in the title is due to Soheyla Feyzbakhsh, who showed that a generic K3 surface X can be geometrically reconstructed from any curve in X of minimal possible degree.
March 8, 2019
Bhargav Narayanan :
3 p.m. in 636 SEO
Abstract
Is there a single sequence of directions that “solves” every maze? The question is trivial for finite mazes,
but becomes far more interesting for infinite mazes. I will speak about various problems that arise from
this seemingly innocuous question, focusing on connections to various “resilience properties” of the
simple random walk.
March 15, 2019
Kathryn Mann :
3 p.m. in 636 SEO
Abstract
The groups Homeo(M) and Diff(M) of homeomorphisms or diffeomorphisms of a manifold M have many striking parallels with finite dimensional Lie groups. In this talk, I'll describe some of these, and explain new work, joint with Lei Chen, that gives an orbit classification theorem and a structure theorem for actions of homeomorphism and diffeomorphism groups on other spaces, analogous to some classical results for actions of locally compact Lie groups. As applications, we answer many concrete questions towards classifying all actions of Diff(M) on other manifolds (many of which are nontrivial, for instance Diff(M) acts naturally on the unit tangent bundle of M...) and resolve several threads in a research program initiated by Ghys. I'll aim to give both a broad overview and several toy applications in the talk.
March 22, 2019
Hans Mueller :
3 p.m. in 636 SEO
Abstract
Samples of random densities and other non-Euclidean data are increasingly encountered in data analysis and meaningful notions of mean, regression and covariance for such data are of statistical interest.
This motivates a general class of regression models that relate responses consisting of random objects in a metric space with Euclidean predictors. In extension of the classical concept of Fréchet means (Fréchet 1948), this leads to conditional Fréchet means, which can be estimated with generalized versions of both global least squares and local weighted least squares regression. These approaches will be illustrated for the special case where the random objects are one-dimensional densities and where one chooses the Wasserstein metric on the space of densities. When data consist of vectors of random densities, the notion of Wasserstein covariance,
defined as an expected inner product of optimal transports, can be used to quantify the dependence of the components of these vectors. Applications include data from demography and brain imaging.
April 5, 2019
Phillip Griffiths :
3 p.m. in 636 SEO
Abstract
Moduli spaces of varieties X are of central interest in algebraic geometry.
For $X$ smooth and of general type the moduli space $M$ exists and has a canonical projective completion $\overline{M}$.
Aside from the case of algebraic curves there are essentially no general results about or examples of the structure of the boundary $\overline{M} \setminus M$.
Hodge theory provides the basic invariant of a complex algebraic variety.
Using Lie theory the space of Hodge structures and its boundary is well understood.
It is therefore natural to use the Hodge- theoretic boundary to study $\overline{M}\setminus M$. This talk will give an informal presentation of this approach together with one result and one application to moduli of a particularly interesting algebraic surface.
*Based on joint work in progress with Mark Green, Radu Laza, and Colleen Robles.
April 12, 2019
Nikhil Srivastava :
3 p.m. in 636 SEO
Abstract
Eigenvalues of random matrices play a central role in many areas of applied mathematics and computer science. Asymptotic random matrix theory has been immensely successful at precisely explaining the limiting spectra of large random matrices with independent entries (or other symmetries). For more general models in finite dimensions, the picture is less crystalline but tools such as the "Matrix Chernoff Bound" give useful coarse bounds on the extreme eigenvalues.
I will describe an object which shares features of both these regimes --- the expected characteristic polynomials of finite random matrices --- and which can be used to show that some of the sharp bounds from the former setting hold with non-negligible probability in the latter. The technique is based on certain interlacing relations between polynomials with all real roots, and is elementary and should be accessible to a general audience.
Based on joint work with Adam Marcus and Daniel Spielman.
April 26, 2019
Richard Samworth :
3 p.m. in 636 SEO
Abstract
When faced with a dataset and a statistical problem of interest, should we propose a statistical model and use that to inform an appropriate algorithm, or dream up a potential algorithm and then seek to justify it? The former is the more traditional statistical approach, but the latter appears to be becoming more popular. I will present an example of a 20th century analysis that falls into the first category, and explain why it may not be as suitable for modern statistical challenges. I'll then discuss a class of algorithms that belong in the second category, namely those that involve data perturbation (e.g. subsampling, random projections, artificial noise, knockoffs,...). As an illustration, I will consider Complementary Pairs Stability Selection for variable selection.
Sept. 6, 2019
Charles Smart :
3 p.m. in 636 SEO
Abstract
Anderson localization is a physical phenomenon in which electron transport is inhibited by the presence of disorder.
The mathematical theory of Anderson localization has a large literature and many important open problems.
I will discuss joint work with Jian Ding in which we establish localization near the edge for the Anderson Bernoulli model on the two dimensional lattice.
Our proof follows the program of Bourgain--Kenig and uses a new unique continuation result inspired by Buhovsky--Logunov--Malinnikova--Sodin.
Sept. 13, 2019
Hao Huang :
3 p.m. in 636 SEO
Abstract
In the $n$-dimensional hypercube graph, one can easily choose half of the vertices such that they induce an empty graph. However, having even just one more vertex would cause the induced subgraph to contain a vertex of degree at least $\sqrt{n}$. This result is best possible, and improves a logarithmic lower bound shown by Chung, Furedi, Graham and Seymour in 1988. In this talk we will discuss a very short algebraic proof of it.
As a direct corollary of this purely combinatorial result, the sensitivity and degree of every boolean function are polynomially related. This solves an outstanding foundational problem in theoretical computer science, the Sensitivity Conjecture of Nisan and Szegedy.
Sept. 20, 2019
Michelle Chu & Marcus Michelen :
3 p.m. in 636 SEO
Abstract
This is the first of two colloquia highlighting our new Research Assistant Professors. Titles/abstracts are as follows:
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(Michelle) Title: Arithmetic hyperbolic 3-manifolds
Abstract: The study of virtual properties of 3-manifolds groups has played a key role in major recent developments in 3-manifold topology. In this talk I will motivate and introduce the study of arithmetic hyperbolic manifolds and discuss some recent results on quantifying their virtual properties.
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(Marcus) Title: Zeros of Polynomials and Central Limit Theorems
Abstract: Let $f$ be a polynomial with non-negative real coefficients. Pemantle conjectured that if $f$ has no roots close to $1 \in \mathbb{C}$, then the coefficients of $f$ roughly trace out a Gaussian bell curve. In the language of probability, this says that the random variable $X$ defined by $$\frac{f(z)}{f(1)} = \sum_k \mathbb{P}(X = k)z^k $$
is close to a normal variable provided the variance of $X$ is large and $f$ has no roots near $1$. I will discuss a complete resolution of this conjecture in a strong quantitative form. Additionally, if $f$ has no roots with small argument, then $X$ must be approximately normal, again in a sharp quantitative form. Time permitting, I will discuss an application of these results to probability and combinatorics. This talk is based on joint work with Julian Sahasrabudhe.
Sept. 27, 2019
Wei Ho :
3 p.m. in 636 SEO
Abstract
Elliptic curves are fundamental and well-studied objects in arithmetic geometry. However, much is still not known about many basic properties, such as the number of rational points on a "random" elliptic curve. We will discuss some conjectures and theorems about this "arithmetic statistics" problem, and then show how they can be applied to answer a related question about the number of integral points on elliptic curves over Q. In particular, we show that the second moment (and the average) for the number of integral points on elliptic curves over Q is bounded (joint work with Levent Alpoge).
Oct. 4, 2019
Filippo Calderoni & Daniel Lear Claveras :
3 p.m. in 636 SEO
Abstract
This is the second of two colloquia highlighting our new Research Assistant Professors. Titles/abstracts are as follows:
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(Filippo) Title: Recent results on permutation groups
Abstract: The automorphism groups of countable homogeneous structures
are natural examples of separable and completely metrizable
topological groups. The richness of their topological properties have
recently brought to light a crucial interplay between Fraïssé
amalgamation theory and other areas of mathematics such as Ramsey
theory and topological dynamics. Moreover they have been studied
extensively as permutation groups.
In this talk we focus on the latter aspect. We will discuss how
certain model theoretic properties are used to analyze the normal
subgroup structure of a large class of those groups. In particular,
will see that if $M$ is the order expansion of the Fraïssé limit of a
free, transitive and nontrivial amalgamation class, then $Aut(M)$ is
simple. This is joint work with Kwiatkowska and Tent.
------
(Daniel) Title: Stability Near Hydrostatic Equilibrium in Fluid Mechanics
Abstract: A fluid is said to be in hydrostatic equilibrium when it is at rest. Then the forces acting on it must balance it.
A natural question therefore arises: What happens if our initial data is close to an hydrostatic equilibrium solution?
The field of hydrodynamic stability has a long history starting in the 19th century. For us, the basic problem is to consider a perturbation of the hydrostatic equilibrium, in which case the fluid must start to move, and to study the long-time behavior of the solution.
Oct. 11, 2019
Moon Duchin :
3 p.m. in 636 SEO
Abstract
Random walks are a fundamental area of study in pure math, and they undergird the technique called Markov chain Monte Carlo (MCMC) that drives a huge range of statistical applications in science, engineering, and industry. I'll explain how MCMC found its way into the redistricting conversation, and I'll introduce a new Markov chain called "Recombination"—a graph partition chain designed for redistricting—and contrast that with the standard "Flip" chain. Along the way, I hope to explain some tradeoffs and debates in the use of math models in politics, policy, and law.
<br>
<b>There will be a Q&A session aimed at graduate students from 4:30 to 5:30 in 636 SEO.</b>
Oct. 18, 2019
Robert Haslhofer :
3 p.m. in 636 SEO
Abstract
I'll describe a new link between geometry and probability. First, I'll give a basic introduction to Ricci curvature, the Einstein equations and Hamilton's Ricci flow. Next, I'll give an introduction to Brownian motion in Euclidean space and on manifolds, and integration by parts on path space. Finally, I'll present joint work with Aaron Naber where we discovered an infinite dimensional Bochner formula for martingales on path space, which vastly generalizes the classical Bochner formula for the heat flow on manifolds. Using these ideas, we can make sense of solutions of the Einstein equations and the Ricci flow in the setting of singular spaces, which solves a long-standing open problem.
Oct. 25, 2019
Philippe LeFloch :
3 p.m. in 636 SEO
Abstract
I will overview recent advances on the mathematical theory of wave propagation in fluids, especially problems involving several scales and nonlinear interactions between shocks, gravitational waves, and phase interfaces. Understanding the global dynamics of complex fluid flows requires insights and methods from the fields of continuum physics, geometric analysis (curvature operators on a Riemannian manifold), partial differential equations (Euler equations, Einstein equations), and scientific computation (finite volume methods, mesh-free algorithms). Blog: philippelefloch.org
Nov. 1, 2019
David Fisher :
3 p.m. in 636 SEO
Abstract
This talk is part of the Midwest Dynamical Systems Conference - http://homepages.math.uic.edu/~hurder/mwds2019/
Totally geodesic submanifolds play an important role in the theory of hyperbolic manifolds.
I will discuss a new rigidity theorem in this context: if a finite volume hyperbolic manifold M contains infinitely many closed totally geodesic hypersurfaces, then M is arithmetic. This answers a question asked by Reid and McMullen. I will explain why it is natural to think of arithmetic manifolds as rare or special in this context. I will also discuss a variant of the theorem for closed totally geodesic submanifolds of higher codimension and also an analogue where M is complex hyperbolic.
I hope to give some ideas of the proofs. The proof in the real hyperbolic case is a combination of homogeneous dynamics with a superrigidity theorem also proven by dynamical methods. The proof in the complex hyperbolic case is more complicated. In addition to using those tools, it draws on the theory of Higgs bundles and also on a theorem about incidence geometry proven by Pozzetti in her study of maximal representations.
This is joint work with Bader, Miller and Stover.
Nov. 8, 2019
Vlad Vicol :
3 p.m. in 636 SEO
Abstract
The mathematical analysis of shock formation for the Euler equations has a long and rich history, particularly in the case of one space dimension, which allows the full power of the method of characteristics to be employed. The first proof of shock formation for the compressible Euler equations in the multi-dimensional setting was given by Christodoulou-Miao ('14) with the restriction to irrotational flow.
In a recent joint work with T. Buckmaster (Princeton) and S. Shkoller (UC Davis) we have provided an elementary constructive proof of shock formation from smooth initial datum of finite energy, with no vacuum regions, and with order unity vorticity.
The blowup time and location can be explicitly computed and the solution at the blowup time is precisely of cusp-type, with precisely Holder $C^{1/3}$ regularity.
Nov. 15, 2019
Anand Pillay :
3 p.m. in 636 SEO
Abstract
I will talk about some model theoretic methods with a "nonstandard" flavour.
I will first mention a second generation proof of function field Mordell-Lang in positive characteristic (2016). And then discuss recent and current work on arithmetic regularity lemmas from combinatorics.
Nov. 22, 2019
Roshan Joseph :
1:30 p.m. in 636 SEO
Abstract
This talk presents a novel method called support points, which can be used for optimal and model-free subsampling of big data. This method has important applications to many practical problems in statistics and machine learning, particularly when the available data is plentiful and high-dimensional, but the processing of such data is expensive due to computation or storage costs. We also propose an extension of the method called Projected Support Points to deal with high dimensional data, which ensures that the data is well-reduced on low-dimensional projections of the data space.
Feb. 21, 2020
Colette Guillope :
3 p.m. in 636 SEO
Abstract
In this talk I shall describe the situation of women in science, and talk in particular about the Gender Gap in Science.
A worldwide 3-year project « A Global Approach to the Gender Gap in Mathematical, Computing and Natural Sciences: How to Measure It, How to Reduce It? » has been funded by ISC (International Science Council). Eleven scientific unions and worldwide organisations, including IMU (International Mathematics Union) were partners in organising the project along three directions: a global survey of scientists, women and men, with more than 32,000 answers; an investigation of gender patterns in millions of scientific publications; and the setting-up of a best-practice database to encourage girls and young women to enter STEM fields.
March 13, 2020
Online Open House Talks :
3 p.m. in Online
April 3, 2020
Jesse Wolfson :
3 p.m. in 612 SEO
April 10, 2020
Laszlo Lempert (CANCELLED) :
3 p.m. in 636 SEO
April 17, 2020
Jacob Fox :
3 p.m. in 636 SEO
April 24, 2020
Bob Kohn :
3 p.m. in 636 SEO
May 1, 2020
Hugh Woodin :
3 p.m. in 636 SEO
Sept. 18, 2020
Anudeep Kumar, Aditya Potukuchi, Geoffrey Smith :
3 p.m. in Zoom
Abstract
This colloquium will highlight our new Research Assistant Professors. Titles/Abstracts are as follows:
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Geoffrey Smith -- Title: Algebraic geometry in positive characteristic
Abstract: Algebraic geometers tend to like working in characteristic 0, and with good reason; a lot of the most basic tools we work with fall apart in positive characteristic. I'll give a couple examples of this happening and of how the extra challenges of positive characteristic have impacted my work.
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Anudeep Kumar -- Title: Singularities and global solutions in the Schrödinger-Hartree equation
Abstract: In 1922, Louis de Broglie proposed wave-particle duality and introduced the idea of matter waves. In 1925, Erwin Schrödinger, proposed a wave equation for de Broglie’s matter waves. The Schrödinger equation is described using de Broglie’s matter wave, which takes the wave function, and describes its quantum state over time.
We consider a nonlinear Schrödinger type equation with nonlocal nonlinearity, of a convolution type, called the generalized Hartree (gHartree) equation. In the gHartree equation, the influence on the behavior of the solutions is global as opposed to the case of local (power type) nonlinearities. In the inter-critical regime, we first obtain a dichotomy for global (scattering) vs finite time (blow-up) existing solutions exhibiting two methods of obtaining scattering: one via Kenig-Merle concentration - compactness and another one is using Dodson-Murphy approach via Morawetz estimate and Tao's scattering criteria. Next, we investigate stable singularity formations in the mass-critical gHartree equation, and rigorously prove a stable blow-up formation in dimension 3.
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Aditya Potukuchi -- Title: On the list recoverability of randomly punctured codes
Abstract: An error-correcting code of block length n is said to be (l,L) list recoverable if for any sets A_1, A_2,...,A_n of the alphabet, each for size at most l, the number of codewords in A_1 x A_2 x ... x A_n is at most L. This simple combinatorial property turns out to be useful in many contexts in Algorithms and Complexity Theory. I will talk about the list recovery properties of randomly punctured codes. In particular, we will see that these codes typically have list recoverability better than what is guaranteed by the Johnson bound. Joint work with Ben Lund (IBS).
Sept. 25, 2020
Rupert Frank :
3 p.m. in Zoom
Abstract
The liquid drop model is an isoperimetric problem with a competing non-local term. It was originally introduced in the nuclear physics literature in 1930 and has received a lot of attention recently as an interesting problem in the calculus of variations. We discuss some new results and open problems. We show how the insights from this problem allowed us to prove the ionization conjecture in a certain model for an atom in density functional theory.
Nov. 20, 2020
Tristan Buckmaster :
3 p.m. in Zoom
Abstract
I will talk about recent work with Steve Shkoller, and Vlad Vicol, regarding shock wave formation for the compressible Euler equations.
Jan. 29, 2021
Dhruv Mubayi :
3 p.m. in Zoom
Abstract
Ramsey theory studies the paradigm that every sufficiently large system contains a well-structured subsystem. Within graph theory, this translates to the following statement: for every positive integer s, there exists a positive integer n so that for every partition of the edges of the complete graph on n vertices into two classes, one of the classes must contain a complete subgraph on s vertices. Beginning with the foundational work of Ramsey in 1928, the main question in the area is to determine the smallest n that satisfies this property.
For many decades, randomness has proved to be the central idea used to address this question. Very recently, we proved a theorem which suggests that ``pseudo-randomness” and not complete randomness may in fact be a more important concept in this area. This new connection opens the possibility to use tools from algebra, geometry, and number theory to address the most fundamental questions in Ramsey theory. This is joint work with Jacques Verstraete.
March 5, 2021
Joseph Teran :
3 p.m. in Zoom
Abstract
Hyperelastic constitutive models describe a wide range of materials. Examples include biomechanical soft tissues like muscle, tendon, skin etc. Elastoplastic materials consisting of a hyperelastic constitutive model combined with a notion of stress constraint (or feasible stress region) describe an even wider range of materials. A very interesting class of these models arises from frictional contact considerations. Examples include granular materials like sand and snow. I will present recent models developed for novel applications including frictional contact for membrane and thin shell simulation, ductile fracture and baking of breads and cookies. I will also present novel Material Point Methods (MPM) used for approximating the governing equations.
April 2, 2021
Uri Bader :
3 p.m. in Zoom
Abstract
Compact hyperbolic manifolds are very interesting geometric objects.
Maybe surprisingly, they are also interesting from an algebraic point of view:
They are completely determined by their fundamental groups (this is Mostow's Theorem),
which is naturally a subgroup of the rational valued invertible matrices in some dimension, GL_n(Q).
When the fundamental group essentially consists of the integer points of some algebraic subgroup of GL_n we say that the manifold is arithmetic.
A question arises: is there a simple geometric criterion for arithmeticity of hyperbolic manifolds?
Such a criterion, relating arithmeticity to the existence of totally geodesic submanifolds, was conjectured by Reid and by McMullen.
In a recent work with Fisher, Miller and Stover we proved this conjecture.
Our proof is based on the theory of AREA, namely Algebraic Representation of Ergodic Actions, which Alex Furman and I have developed in recent years.
In this colloquium talk I will survey the subject in a colloquial manner.
April 9, 2021
Thomas Nikolaus :
3 p.m. in Zoom
Abstract
We explain how to compute the stable cohomology of the groups O(n,n;Z) and Sp(n, Z). Here stable means that n tends to infinity. This is joint work with Hebestreit and Land and is a consequence of recent results about Grothendieck-Witt theory.
April 23, 2021
Rahul Pandharipande :
3 p.m. in Zoom
Abstract
Hurwitz's paper "Ueber die Anzahl der Riemann'schen Flächen mit gegebenen Verzweigungspunkten" (1901) started the study of the enumeration of branched coverings of the Riemann sphere. Though more than a century has passed now, there have been many recent developments in the subject that Hurwitz opened. I will explain new results and perspectives on Hurwitz numbers, Hurwitz moduli spaces, and related constructions concerning meromorphic differentials.
April 30, 2021
Jerry Bona :
3 p.m. in Zoom
Abstract
Mathematics has a long and distinguished history of being
useful in understanding problems arising in oceanography.
There are many areas of oceanography upon which mathematical
analysis of one sort or another has been able to cast light.
There are far too many to survey them all in an hour, so I
have chosen two topics to discuss in a little detail. One of
these is the propagation of internal waves and the other
is the formation of rogue waves - a.k.a. freak waves or
giant waves.
Sept. 17, 2021
Jacob Tsimerman :
3 p.m. in Zoom
Abstract
(Joint with Jonathan Pila and Ananth Shankar) Shimura varieties are objects which are at the heart of arithmetic geometry, yet they start life as purely complex analytic objects. Many Shimura varieties are moduli spaces---mostly relating to abelian varieties---and this makes their arithmetic much more accessible. However, for the ones that have no moduli interpretation, we only see shadows of these structures as Galois representations, variations of Hodge structures, modular forms, etc. We explain how recent advances---particularly in relative p-adic Hodge theory---allow us to make precise arithmetic statements about Shimura varieties that were previously inaccessible. Primarily, we explain how to create a canonical height for arbitrary Shimura varieties, analogous to the Faltings height for the Siegel modular variety.
As our chief application, we explain how to use this theory to complete the proof of the Andre--Oort conjecture. While this conjecture can be made purely in the analytic world---describing the Zariski distribution of special points---its proof involves much algebraic and analytic number theory relying heavily on point counting results of Pila-Wilkie and recent improvements of Binyamini. We will try to present an overview of all the ingredients that go into the proof.
Nov. 12, 2021
Richard D. James :
3 p.m. in 636 SEO
Abstract
We begin with a general introduction to our work on “objective structures” with a focus on the relation between symmetry, invariance and structure,
and applications to the periodic table, origami design and nanostructures. We then concentrate on Maxwell’s equations. We find solutions of
Maxwell’s equations that are the precise analog of plane waves, but in the case that the translation group is replaced by the (largest)
Abelian helical group. These waves display constructive/destructive interference with helical atomic structures, in the same way that plane
waves interact with crystals. We show how the resulting far-field pattern can be used for structure determination, and we test the idea
theoretically on the Pf1 virus from the Protein Data Bank. The underlying mathematical idea of this and our related work is: the structure
of interest is the orbit of a group, and this group is an invariance group of the differential equations.
Nov. 23, 2021
Departmental Hold :
3 p.m. in Zoom
Nov. 30, 2021
Departmental Hold :
3 p.m. in Zoom
Dec. 1, 2021
Departmental Hold :
3 p.m. in Zoom
Dec. 2, 2021
Departmental Hold :
3 p.m. in Zoom
Dec. 3, 2021
Departmental Hold :
3 p.m. in Zoom
Dec. 6, 2021
Departmental Hold :
3 p.m. in Zoom
Dec. 7, 2021
Departmental Hold :
3 p.m. in Zoom
Jan. 5, 2022
Department Hold :
4:15 p.m. in Zoom
Jan. 6, 2022
Department Hold :
4:15 p.m. in Zoom
Departmental Hold :
3 p.m. in Zoom
Jan. 7, 2022
Department Hold :
4:15 p.m. in Zoom
Departmental Hold :
3 p.m. in Zoom
Jan. 10, 2022
Department Hold :
4:15 p.m. in Zoom
Jan. 11, 2022
Departmental Hold :
3 p.m. in Zoom
Department Hold :
4:15 p.m. in Zoom
Jan. 12, 2022
Department Hold :
4:15 p.m. in Zoom
Departmental Hold :
3 p.m. in Zoom
Jan. 13, 2022
Department Hold :
4:15 p.m. in Zoom
Departmental Hold :
3 p.m. in Zoom
Jan. 14, 2022
Department Hold :
4:15 p.m. in Zoom
Departmental Hold :
3 p.m. in Zoom
Jan. 18, 2022
Departmental Hold :
3 p.m. in Zoom
Jan. 19, 2022
Department Hold :
4:15 p.m. in Zoom
Department Hold :
3 p.m. in Zoom
Jan. 20, 2022
Department Hold :
4:15 p.m. in Zoom
March 18, 2022
Oscar García-Prada :
3 p.m. in 636 SEO
Abstract
Given a closed orientable surface S and a Lie group G, one can consider the set of equivalence classes of representationsof the fundamental group of S in G, which is often referred as the G-character variety of the fundamental group of S. Choosing a complex structure on S, one can identify the elements in the G-character variety with certain holomorphic objects on the corresponding Riemann surface. If G is the circle, this gives the identification of the character variety with the Jacobian of the Riemann surface --- correspondence that goes back to the 19th century. In this talk we will focus on the case in which G is a non-abelian non-compact semisimple Lie group. In this situation, the non-abelian Hodge correspondence identifies the G-character variety with the space of some holomophic objects on the Riemann surface known as G-Higgs bundles. Using this correspondence, we give a classification of the simple Lie groups G for which the G-character variety has components that generalize the Teichmüller space of S parameterizing complex structures on S, regarded as a topological component of the character variety for G=PSL(2,R).
April 8, 2022
Shiv Karunakaran :
3 p.m. in Zoom
Abstract
An "introduction to mathematical reasoning" or “Transition to mathematical proof” course is widely offered as an essential part of the undergraduate mathematics curriculum at most institutions of high learning. The importance of such a course cannot be overstated, as the concepts and skills learned in this course are foundational to any subsequent mathematics course. However, research suggests that undergraduate students continue to struggle understanding the function of mathematical proof and proving. This talk will report on the journey to a currently NSF-funded project examining the impact of a student-centered Intro to Proof course. I will detail both the empirical research work and the course design work that went into informing and influencing the current project.
Sept. 2, 2022
Tom Benhamou and Benjamin Call :
3 p.m. in 636 SEO
Abstract
Singular cardinals, the forcing method and Prikry-type forcing
In the first part of this talk, we shall present an overview and give motivation for the forcing method, which is a method to produce models of set
theory. Next, we shall discuss the problems of the generalized continuum hypothesis for singular cardinals, and the problem of preserving cardinals and cofinalities.
Finally, we will introduce the main tool to work on with singular cardinals- Prikry
type forcings- which produces models where cardinals are preserved and cofinality
of some cardinals changes. More specifically, we will focus on Prikry forcing
and Magidor forcing and present recent results about those forcings.
The audience is assumed to be familiar with basic concept of set theory and
logic, such as models, ordinals and cardinals.
Unique Equilibrium States for Geodesic Flows
Equilibrium states (which are simply special invariant measures) for the geodesic flow on a compact, negatively curved manifolds, are well known to have many extremely nice properties. A natural question is what we can say when we weaken our assumptions on the geometry of our setting, for instance to nonpositively curved manifolds, or CAT(0) spaces. In this talk, we will provide a brief introduction to the theory of equilibrium states and provide an overview of recent developments in showing uniqueness and mixing properties of these measures beyond the setting of negative curvature. Finally, time permitting, we will give an idea of one of the main techniques used in recent work to handle the case of the geodesic flow on translation surfaces.
Sept. 9, 2022
Nolan Schock and Liet Vo :
3 p.m. in TBA
Abstract
Generalizations of Combinatorial Hodge Theory and Gromov-Witten Theory
June Huh was recently awarded a Fields Medal for his work on the development of ``Combinatorial Hodge Theory,'' allowing one to use techniques of algebraic geometry to solve a number of long-standing conjectures in combinatorics. Gromov-Witten Theory was developed in the 1990s as a tool for counting curves on algebraic varieties using ideas inspired by physics. I will describe how one might ambitiously hope to use generalizations of Combinatorial Hodge Theory to develop higher-dimensional generalizations of Gromov-Witten Theory. (No prior knowledge of either of these topics will be necessary.)
Numerical methods for stochastic Stokes and Navier-Stokes equations
Navier-Stokes and Stokes equations are one of the most well-known equations in fluid
mechanics because of their broad applications. In this colloquium, I will introduce the stochastic
versions of these equations. It is well-known that the stochastic Navier-Stokes and Stokes equations
are used for a better understanding of turbulence and also thermodynamic fluctuations present in
fluid flows. We will focus on the numerical methods to solve the stochastic Navier-Stokes and Stokes
equations. I will introduce the two popular methods that are Euler-Maruyama-mixed finite element
method and Chorin projection method for solving these equations. For each case, error estimates
are also addressed and analyzed.
Oct. 14, 2022
Artem Chernikov :
3 p.m. in Library Conference Room 1-470
Abstract
Finite VC-dimension, a combinatorial property of families of sets, was discovered simultaneously by Vapnik and Chervonenkis in probabilistic learning theory, and by Shelah in model theory (where it is called NIP). It plays an important role in several areas including machine learning, combinatorics, mathematical logic, functional analysis and topological dynamics. We develop aspects of higher-order VC-theory, establishing a higher arity generalization of the epsilon-net theorem for sets (and functions) with bounded VC_k-dimension. As an application, we obtain a strong version of Szemerédi's regularity lemma for hypergraphs omitting a fixed finite k-partite k-hypergraph. Joint work with Henry Towsner.
Oct. 28, 2022
Nikhil Bansal :
3 p.m. in 636 SEO
Abstract
In the classical two-choice process for assigning balls into bins, each ball chooses
two bins uniformly at random and is placed greedily in the least loaded of the two
bins. This power-of-two-choices paradigm has been highly influential and leads
to substantially better load balancing than a random assignment of balls into bins.
Somewhat surprisingly, the greedy strategy turns out to be quite sub-optimal
for some natural generalizations. One such setting is the graphical process
where the bins correspond to the vertices of a graph G, and at any time
a random edge is picked and a ball must be assigned to one of its end-points.
Another setting is where the balls can also be deleted arbitrarily by an oblivious
adversary. In this talk we will see why the greedy strategy can perform poorly, and
I will describe other strategies for these settings that are close to optimal.
Based on joint works with Ohad Feldheim and William Kuszmaul.
Nov. 4, 2022
Huy Tuan Pham :
3 p.m. in 636 SEO
Abstract
The threshold of an increasing graph property is the density at which a random graph transitions from unlikely satisfying to likely satisfying the property. Kahn and Kalai conjectured that this threshold is always within a logarithmic factor of the expectation threshold, a natural lower bound to the threshold which is often much easier to compute. In probabilistic combinatorics and random graph theory, the Kahn—Kalai conjecture directly implies a number of difficult results, such as Shamir’s problem on hypergraph matchings, or the threshold for containing a bounded degree spanning tree. I will discuss recent joint work with Jinyoung Park that resolves the Kahn—Kalai conjecture. Our proof of the Kahn—Kalai conjecture is closely related to the resolution (in joint work with Jinyoung Park) of a conjecture of Talagrand on extreme events of suprema of certain stochastic processes driven by sparse Bernoulli random variables (known as selector processes), and a question of Talagrand on suprema of general positive empirical processes. Given recent advances on chaining and the resolution of the (generalized) Bernoulli conjecture, these results give the first steps towards Talagrand’s last ``Unfulfilled dreams’’ in the study of suprema of general empirical processes. These problems turn out to be intimately related to the sunflower conjecture, an important open problem in extremal combinatorics which has surprising connections to theoretical computer science. Time permitting, I will discuss the connections to sunflowers in various settings.
Nov. 11, 2022
Alexander Barvinok :
3 p.m. in 636 SEO
Abstract
On a few examples, such as the permanent of a matrix (partition function for a system of bosons), matching polynomial of a graph (partition function in a monomer-dimer system) and the independence polynomial of a graph (partition function in the hard core model), we illustrate how the computational complexity of approximation is related to complex zeros (even if we want to approximate in a real domain) and to the phase transition in the Lee - Yang sense. The idea is roughly as follows: a combinatorially defined multivariate polynomial can be efficiently approximated in a complex domain, if to does not have zeros in a slightly larger domain.
Nov. 16, 2022
Leonardo Coregliano :
3 p.m. in 636 SEO
Abstract
The theory of graph quasirandomness studies graphs that "look like" samples of the Erdős--Rényi
random graph $G_{n,p}$. The upshot of the theory is that several ways of comparing a sequence with
the random graph turn out to be equivalent. For example, two equivalent characterizations of
quasirandom graph sequences is as those that are uniquely colorable or uniquely orderable, that is,
all colorings (orderings, respectively) of the graphs "look approximately the same". Since then,
generalizations of the theory of quasirandomness have been obtained in an ad hoc way for several
different combinatorial objects, such as digraphs, tournaments, hypergraphs, permutations, etc.
The theory of graph quasirandomness was one of the main motivations for the development of the
theory of limits of graph sequences, graphons. Similarly to quasirandomness, generalizations of
graphons were obtained in an ad hoc way for several combinatorial objects. However, differently from
quasirandomness, for the theory of limits of combinatorial objects (continuous combinatorics), the
theories of flag algebras and theons developed limits of arbitrary combinatorial objects in a
uniform and general framework.
In this talk, I will present the theory of natural quasirandomness, which provides a uniform and
general treatment of quasirandomness in the same setting as continuous combinatorics. The talk will
focus on the first main result of natural quasirandomness: the equivalence of unique colorability
and unique orderability for arbitrary combinatorial objects. Although the theory heavily uses the
language and techniques of continuous combinatorics from both flag algebras and theons, no
familiarity with the topic is required as I will also briefly cover all definitions and theorems
necessary.
This talk is based on joint work with Alexander A. Razborov.
Nov. 17, 2022
Beibei Liu :
3 p.m. in 636 SEO
Abstract
The critical exponent is an important numerical invariant of discrete isometry groups acting on negatively curved Hadamard manifolds, Gromov hyperbolic spaces, and higher-rank symmetric spaces. In this talk, I will focus on discrete isometry groups acting on hyperbolic spaces, which is a family of important examples of these three types of spaces. In particular, I will explain how the numerical invariant is closely related to geometry, dynamics, and representation of the group action on the hyperbolic space.
Nov. 18, 2022
Bhargav Narayanan :
2 p.m. in 636 SEO
Abstract
A basic problem at the intersection of probability and combinatorics is the Littlewood-Offord anti
concentration problem: given real numbers a_1, ... , a_n, what is the largest possible point probability
of the random sum a_1 X_1 +... + a_n X_n for iid Bernoulli random variables X_1, ... , X_n?
Several variants of this problem, involving additional arithmetic constraints on the numbers
a_1, ... , a_n, have proved to both be deep and widely applicable; two notable examples of such
variants include the Sarkozy-Szemeredi theorem (resolving the Erdos-Moser problem) and Halasz's
theorem. A few years ago, it became evident to me that all of these arithmetic results are in fact
specialisations of a more abstract, purely combinatorial phenomenon. In this talk, I will take the scenic
route to the recent proof of such an abstract result, regarding the density of “antichain codes” in the
Boolean hypercube, surveying the history of these problems and some of the many applications
along the way.
Nov. 28, 2022
Dori Bejleri :
3 p.m. in 636 SEO
Abstract
It has been said that working with non-compact spaces is like trying to hold change in your pocket with a hole in it. One of the central examples of non-compact spaces in algebraic geometry are moduli spaces. Broadly speaking, the points of a moduli space represent equivalence classes of algebraic varieties of a given type, and its geometry reflects the ways these varieties deform in algebraic families. The classification of algebraic varieties of a given type is tantamount to understanding the geometry of the corresponding moduli space. The goal of this talk is to discuss recent progress on compactifying moduli spaces of higher dimensional varieties, focusing on the interplay between compactifications of moduli spaces and singular degenerations of the objects they classify.
Nov. 29, 2022
Alexander Dunlap :
3 p.m. in 636 SEO
Abstract
A pervading question in the study of stochastic PDE is how small-scale random forcing in an equation combines to create nontrivial statistical behavior on large spatial and temporal scales. I will discuss recent progress on this topic for several related stochastic PDEs - stochastic heat, KPZ, and Burgers equations - and some of their generalizations. These equations are (conjecturally) universal models of physical processes such as a polymer in a random environment, the growth of a random interface, branching Brownian motion, and the voter model. The large-scale behavior of solutions on large scales is complex, and in particular depends qualitatively on the dimension of the space. I will describe the phenomenology, and then describe several results and challenging problems on invariant measures, growth exponents, and limiting distributions.
Dec. 1, 2022
Hana Jia Kong :
3 p.m. in 636 SEO
Abstract
For the past 90 years, a fundamental question in classical homotopy theory is to understand the stable homotopy groups of spheres. The most modern method to study these groups is to compare them with the ``motivic stable homotopy groups of spheres". Motivic homotopy theory has its roots in algebraic geometry. As a result of the recent advances, there is a reintegration of algebraic topology and algebraic geometry, with close connections to equivariant homotopy theory and number theory.
In this talk, I will introduce the classical and motivic stable homotopy categories and the connections between the two. I will then talk about the rich properties and extra structures that are present in the motivic stable homotopy category. The presence of these extra structures gives new computational tools that dramatically improve our understanding of the classical stable homotopy groups. Moreover, the flow of information can be reversed as well, producing new results in motivic stable homotopy theory for general fields.
Dec. 2, 2022
Philip Engel :
3 p.m. in 636 SEO
Abstract
I will discuss some ways that triangulations of the two-dimensional sphere connect to various areas of mathematics: (1) how modular forms and the Siegel-Weil formula relate to the enumeration of buckyballs, (2) how representation theory of the symmetric group arises in generalizing these enumerative results, and (3) how triangulated affine structures arise when compactifying moduli spaces of K3 surfaces.
Dec. 5, 2022
Ilias Zadik :
3 p.m. in 636 SEO
Abstract
The mathematical analysis of high dimensional inference models has been a topic of intense research activity over the last years. This field of study has created a plethora of exciting, fruitful and, sometimes, unexpected interactions between the fields of mathematics, computer science, statistics and statistical physics.
In the first part of this talk I will present to you a recent result on the surprising failure of the Metropolis process for the planted clique model, a well-studied random graph inference setting. Our proof is based on leveraging ``bottleneck’’ methods inspired by statistical physics techniques. Notably, our work resolved a well-known open question posed by Jerrum in 1992 and our proposed technique has been leveraged successfully in several other inference settings. In the second part of the talk, I will share with you some new results on how inference techniques can offer alternative proofs of state-of-the-art results in combinatorics, including for example the celebrated fractional Kahn-Kalai conjecture.
Dec. 7, 2022
Rachel Greenfeld :
3 p.m. in 636 SEO
Abstract
Translational tiling is a covering of a space (e.g., Euclidean space) using translated copies of a building block, called a "tile'', without any positive measure overlaps. What are the possible ways that a space can be tiled?
One of the most well known conjectures in this area is the periodic tiling conjecture. It asserts that any tile of Euclidean space can tile the space periodically. This conjecture was posed 35 years ago and has been intensively studied over the years. In a joint work with Terence Tao, we disprove the periodic tiling conjecture in high dimensions. In the talk, I will motivate this result and discuss our proof.
Dec. 8, 2022
Xiaoyu He :
3 p.m. in 636 SEO
Abstract
wo of the most influential theorems in discrete mathematics state, respectively, that diagonal Ramsey numbers grow exponentially and that error-correcting codes for noisy channels exist up to the information limit. The former, proved by Erdős in 1947 using random graphs, led to the development of the probabilistic method in combinatorics. The latter, proved by Shannon in 1948 using random codes, is one of the founding results of coding theory. Since then, the probabilistic method has been a cornerstone in the development of both Ramsey theory and coding theory. In this talk, we give an overview of the important applications of the probabilistic method in these two parallel but interconnected worlds. We then present new results on Ramsey numbers of graphs and hypergraphs and codes correcting deletion errors, all based on probabilistic ideas.
Dec. 9, 2022
Sami Davies :
3 p.m. in 636 SEO
Abstract
In theoretical computer science, algorithm design and analysis is centered around worst-case optimization, where results focus on provable guarantees for how well an algorithm performs on any input. While worst-case analysis has led to deep structural insights into the problems at hand, we lose much of the nuance behind algorithm performance. Instances forcing the worst-case performance of the algorithm may not appear in practice, do not occur under reasonable assumptions on the distribution of problem inputs, or are avoidable if the algorithm has additional information about the input. In this talk, I'll discuss how we can bridge the gap between theory and practice by using frameworks that go beyond worst-case analysis in order to (1) develop a more complete theoretical understanding of a problem's difficulty, and (2) provide performance guarantees that are representative of what happens in practice. I'll demonstrate this by discussing work on the 3-Coloring problem and the Max Flow problem.
Feb. 20, 2023
Gabriel Goldberg :
3 p.m. in 636 SEO
Abstract
One of the main goals of logic is to classify mathematical theories according to their strength. Since Cohen's 1964 proof of the independence of the continuum hypothesis, set theorists have discovered a staggering array of mutually incompatible set theories, and so such a classification may initially seem impossible. But in fact, it seems that each of these theories corresponds to a level of a single linear hierarchy of theories: the large cardinal hierarchy. These are the theories formed from the standard ZFC axioms by adding generalizations of the axiom of infinity, or large cardinal axioms. Whether this correspondence extends to the strongest known theories is open, and this talk will discuss how this question is related to canonical models of set theory and large cardinal axioms so strong they are inconsistent with the axiom of choice.
Feb. 24, 2023
Fernando Granha Jeronimo :
3 p.m. in 636 SEO
Abstract
Expanders are highly connected, but sparse graphs. By combining these
opposing properties, they found a plethora of applications both in theory
and in practice. Codes are objects that enable the protection of data against
corruptions thereby being instrumental in communication and storage. Optimization
underlies much of our understanding of efficient computation. Despite the fundamental
nature of these areas, much is either being discovered or yet to be discovered.
In this talk, we will discuss how synergistic interactions among expansion, codes and
optimization led to recent progress on our understanding of almost optimal codes and
almost optimal expanders. We will also mention future directions along these frontiers.
March 3, 2023
Tsachik Gelander :
3 p.m. in 636 SEO
Abstract
In mathematics in general, it is fruitful to allow randomness. Indeed, it is often easier to deal with random rather than deterministic objects. It seems miraculous, however, when we are able to say more about deterministic objects by treating them as random ones. This idea applies in particular to the theory of locally symmetric manifolds and discrete subgroups of Lie groups.
The theory of invariant random subgroups (IRS), which has been developed quite rapidly during the last decade, has been very fruitful to the study of lattices and their asymptotic invariants. However, restricting to invariant measures limits the scope of problems that one can approach (in particular since the groups involved are highly non-amenable). It was recently realised that the more general notion of stationary random subgroups (SRS) is still very effective and opens paths to deal with questions which were thought to be unreachable.
In the talk I will describe various old and new results concerning arithmetic groups and general locally symmetric manifolds of finite as well as infinite volume that can be proved using `randomness', e.g.:
1. Kazhdan-Margulis minimal covolume theorem.
2. Most hyperbolic manifolds are non-arithmetic (a joint work with A. Levit).
3. Higher rank manifolds of large volume have a large injectivity radius (joint with Abert, Bergeron, Biringer, Nikolov, Raimbault and Samet).
4. Margulis' infinite injectivity radius conjecture: For manifolds of rank at least 2, finite volume is equivalent to bounded injectivity radius (joint with M. Fraczyk).
March 10, 2023
Peter Song :
3 p.m. in 636 SEO
Abstract
Stratification is one statistical principle in data processing to mitigate the underlying population heterogeneity, which is typically handled by clustering when stratum labels are unknown. Many practical problems require post-clustering statistical learning that is challenged by the issue of “double data dipping”, leading to the difficulty of uncertainty quantification. One solution to address this challenge is to perform a simultaneous operation of clustering and estimation in data analyses. Recently we developed a new paradigm of supervised homogeneity pursuit via mixed integer optimization, which provides a conceptually simple and computationally straightforward machinery with the use of suitable constraints in optimization. This novel toolbox has been then applied to solve several real-world problems arising from infectious disease surveillance, influence of environmental exposure to health, and risk factors for aging. Some algorithmic limitations worth future research will be discussed.
March 17, 2023
Philippe LeFloch :
3 p.m. in 636 SEO
Abstract
I will begin with basic notions required to formulate Einstein's field equations and tackle the the global evolution problem for spacetimes. A variety of phenomena are known to arise in this evolution: gravitational waves, dispersion, collapse, formation of singularities, bouncing, etc. While many problems in these directions remain widely open and very challenging, in the past decades major mathematical advances were made for several classes of spacetimes. I will review some of these recent results, namely: (1) nonlinear stability of Minkowski spacetime, (2) collapse of spherically symmetric spacetimes, and (3) scattering in spacetimes admitting a quiescent singularity. This talk is based on joint work with Y. Ma (Xi'an), T.-C. Nguyen (Montpellier), F. Mena (Lisbon), B. LeFloch (Paris), and G. Veneziano (Geneva). Blog: philippelefloch.org
March 31, 2023
Kevin Tucker :
3 p.m. in 636 SEO
Abstract
A common way to picture a singular point of an algebraic variety is by analyzing its link, the intersection of a small embedded sphere with the variety. Heuristically, more complex links correspond to worse singularities. Over the past few years, certain numerical invariants have been used to (roughly) give an upper bound on the size of the fundamental groups of such links for some important classes of singularities in a number of different settings. In positive characteristic, one can use the F-signature -- an invariant giving a quantitative measure of F-regularity and closely related to Kawamata Log Terminal (KLT) singularities in characteristic zero. In this talk, I aim to motivate and discuss a new mixed characteristic analogue of the F-signature defined using the perfectoidization functor of Bhatt-Scholze. As an application, we are able to bound the size of the étale fundamental group of the regular locus of BCM-regular singularities. This is joint work with Hanlin Cai, Seungsu Lee, Linquan Ma and Karl Schwede.
April 7, 2023
Alina Chertok :
3 p.m. in 636 SEO
Abstract
Solutions of many nonlinear PDE systems reveal a multiscale character; thus, their numerical resolution presents some major difficulties. Such problems are typically characterized by a small parameter representing, say, a low Mach or Fraude number. In the limiting regimes, the propagation speeds are very low, and therefore the use of explicit numerical methods would require very restrictive time and space discretization steps due to the CFL condition and restrictions on the smallness of numerical diffusion. This becomes rapidly too costly from a practical point of view, and consequently, numerical solutions for small parameter values may be out of reach. Moreover, standard implicit schemes, which will be uniformly stable, may be inconsistent with the limiting problem and may provide a wrong solution in the zero limits. Thus, designing robust numerical algorithms whose accuracy and efficiency are independent of the values of the small parameter is an important and challenging task. A widely used numerical approach applicable in all-speed regimes is based on asymptotic preserving (AP) numerical methods. AP methods guarantee that for a fixed mesh size and time step, the numerical scheme should automatically transform into a consistent and stable discretization of the limiting system.
In this talk, we will present several AP schemes for Navier–Stokes–Korteweg equation, rotational shallow water equations with Coriolis, and, if time permits, kinetic equations with singular limits.
April 14, 2023
Edmund Harriss :
3 p.m. in 636 SEO
Abstract
We often think about how mathematical ideas can be used to model the world or physical processes within it. It is one of the great motivations for the study of mathematics. What happens when we reverse that? Asking instead how the world can be used to model mathematical ideas, and what we can learn from observing them. An interesting outcome is that this can lead to new ways to interact between mathematics and the world, some surprising like the development of manufacturing techniques. An example is a CNC router this has three axes (labelled naturally X, Y and Z) and it thus provides a model of 3 dimensional linear algebra with a basis. Once that relationship has been established a link between movements of the machine and paths in the linear algebra can be developed. This has received plenty of attention from the manufacturing side, for example in the calculus necessary to correctly accelerate the different axes to create a smooth path. Going into the detail of these ideas will show the creation of the zipform system for curved metal beams that has been used for art sculptures and engineering.
Sept. 1, 2023
Christopher Mahadeo and Eric Jovinelly :
3 p.m. in 636 SEO
Abstract
Christopher Mahadeo
Title: Twisted Higgs bundles and topological recursion
Abstract: I will briefly introduce Higgs bundles and topological recursion, and discuss one way they are related (in both the regular and twisted setting).
Eric Jovinelly
Title: Geometric Manin's Conjecture for Fano Threefolds
Abstract: A famous conjecture of Manin predicts an asymptotic formula for counting the number of rational points of bounded height on a Fano variety defined over a number field. In the 1990s, Batyrev developed a heuristic argument for a version of Manin's Conjecture over finite fields that assumes irreducibility of certain spaces of embedded rational curves. Though Batyrev's heuristics and Manin's initial conjecture are false in general, Geometric Manin’s Conjecture (GMC) translates Batyrev’s heuristic for Manin’s Conjecture to statements about free rational curves on Fano varieties. In this talk, I will first review this translation and motivate the framework of GMC with concrete examples. I will then describe a recent proof of GMC for smooth Fano threefolds over the complex numbers by appealing to relationships between this framework and the Mori structures of Fano threefolds.
Sept. 8, 2023
Scott Mutchnik & Gwyn Moreland :
3 p.m. in 636 SEO
Abstract
Title for Gwyn Moreland: Positive cycles on Hilbert schemes of points
Abstract for Gwyn Moreland: Algebraic geometers are often interested in certain “positive” cones that one can associate to varieties, such as the nef and effective cones. While traditionally studied in the divisor and curves cases, positive cones of cycles of intermediate dimension have recently been the subject of much study. We discuss some results of this kind, and their particular challenges, in the case of Hilbert schemes of points. In particular, we look at the Hilbert scheme of 3 points in P^3, which parametrizes collections of 3 distinct points in 3-space as well as all limits of such subschemes.
Title for Scott Mutchnik: Interactions between the syntactic and the semantic
Abstract for Scott Mutchnik: Model theorists are interested in what the language of logic has to say about mathematical structures. Within model theory, Shelah’s program of classification theory proposes to analyze the universe of mathematical structures according to combinatorial patterns in the syntax of logical theories, at the level of individual formulas. The combinatorics of a theory at the local scale often provides global information on the geometric structure of its models. However, the geometry of general mathematical structures has recently allowed us to solve classical combinatorial problems in the syntax of first-order theories, including the question (asked by Shelah (1999) and Džamonja and Shelah (2004)) of whether NSOP_1 is equal to NSOP_2. We discuss this problem, which makes no explicit mention of anything besides the combinatorics of the formulas themselves, and how our solution reveals the influence of the semantic on the syntactic.
Sept. 15, 2023
Steven Johnson :
3 p.m. in 636 SEO
Abstract
Over the past two decades, an explosion in fabrication capabilities for nano-structured optics has coincided with the development of powerful techniques for "inverse design" — large-scale PDE-constrained optimization, sometimes with millions of degrees of freedom, that reveals surprising irregular structures for a diverse range of devices. Light emission, sensing, communications, and imaging have all begun to exploit rich subwavelength scattering physics with the aid of these techniques. But more than simply applying more computing power, inverse design has pushed mathematical formulations of engineering design to the forefront: tractable reformulations of device objectives, new convexifications to obtain useful bounds on performance, algorithms to incorporate manufacturing constraints, and more. More recently, a new design frontier has emerged in which the desired optical outputs are not even specified in advance. Instead, one targets the information that is to be extracted by a subsequent computational inference (such as regularized regression or even neural networks), and the optical device is co-optimized "end-to-end" along with the image processing/inference to maximize accuracy in the presence of noise. Mathematically, this is a form of stochastic bilevel optimization problem. Building on pioneering on end-to-end design and computational inference for scalar-diffractive surfaces by various groups, the development of fullwave meta-optic end-to-end algorithms allows us to exploit the extreme polarization, direction, and wavelength sensitivity possible in subwavelength structures, in order to obtain radically new designs for problems in hyper-spectral imaging, depth sensing, polarization sensing, thermal imaging, and other challenges on the horizons of optics.
Sept. 22, 2023
Shiji Lyu & Freddy Saia :
3 p.m. in 636 SEO
Abstract
Shiji Lyu Title: Singularities in commutative algebra with a touch of model theory
Shiji Lyu Abstract: In algebraic geometry, the singularities of a variety are usually defined or characterized by its local rings. This gives rise to interesting and important notions and questions in commutative algebra. In this talk, we discuss several results regarding some behaviors of certain notions of singularity, proved with the help of ideas or perspectives from model theory.
Freddy Saia Title: Title: A volcanic approach to CM points on Shimura curves
Freddy Saia Abstract: A CM component of the $\ell$-isogeny graph of elliptic curves has a particular structure, that of an $\ell$-volcano, at least away from certain CM orders. The structure of ``isogeny volcanoes'' has seen much use in the study of CM elliptic curves over finite fields, originating with 1996 thesis work of Kohel. Recent work of Clark--Saia leverages infinite depth versions of these graphs to study moduli of isogenies of CM elliptic curves over $\overline{\mathbb{Q}}$.
We will discuss an analogue of this work for abelian surfaces with quaternionic multiplication. A main result includes an algorithm to compute the $\mathfrak{o}$-CM locus on the Shimura curve $X_0^D(N)$ over $\mathbb{Q}$, for $\mathfrak{o}$ any imaginary quadratic order and $\mathrm{gcd}(D,N) = 1$. As an application, we give an explicit list of pairs $(D,N)$ for which the Shimura curve $X_0^D(N)$ may fail to have a sporadic CM point.
Sept. 29, 2023
Richard Birkett & Daniel Ingebretson :
3 p.m. in 636 SEO
Abstract
Title for Richard Birkett: Dynamics when Zero is Divided by Zero
Abstract for Richard Birkett: What is algebraic dynamics? Generally speaking, this is the iteration of a map defined by rational functions on an algebraic space. In dimension 1, in parallel with the setting of classical complex dynamics, one typically focuses on the Fatou and Julia sets, the loci of order and chaos respectively. On an algebraic variety of any higher dimension, the dynamics of a rational map is more delicate; for instance we cannot define a Fatou-Julia theory that so nicely classifies the dynamical behaviour. Even the way to iterate is unclear: a point in motion may encounter a spot where the expression for the map appears to be zero divided by zero. What happens to the point next? Does this affect the dynamical complexity of the map? I will discuss these problems for rational maps on a complex surface, and describe how to transfer this information to a rather different dynamical system on a one-dimensional non-Archimedean space called the Berkovich projective line. By recovering a Fatou-Julia theory in the latter setting, we gain insight to the behaviour of certain rational maps on surfaces.
Title for Daniel Ingebretson: Hausdorff and packing measure of some linear numeration systems
Abstract for Daniel Ingebretson: The size of a fractal set is often measured using some form of non-integer fractal dimension. For those dimensions defined by a measure, a more subtle question is the measure of the fractal at dimension, and even for simple fractals this is largely unexplored. In this talk, we will discuss a couple of specific cases: the Hausdorff and packing measure of some restricted digit fractal sets arising in numeration systems.
Oct. 6, 2023
Pawel Horodecki :
3 p.m. in 636 SEO
Abstract
Quantum mechanics allows quantum correlations – also called quantum entanglement – that are
stronger than all the correlations we know from our daily lives. Their enigma has already troubled
fathers of Quantum Mechanics. Einstein's ingenious skepticism about this theory gave rise to the
fundamental philosophical question - formalized mathematically by John Bell - about the objective
existence of properties of quantum particles before measurement.
We shall discuss the related Bell inequalities tests from the perspective of randomness and stress
that while quantum mechanical statistics look completely random, it may allow for a contribution of
determinism, if we look at them from the perspective of possible future physical theories. This rises
an interesting problem of certification of randomness and cryptographic security in hypothetical
situations where eavesdropper has a post-quantum power.
The two most natural post-quantum frameworks are the ones of no-signaling boxes and no-
signaling assamblages. We discuss quantum correlations from the perspectives of the two
frameworks. This includes on the one hand some no-go theorems and on the other hand some
positive results concerning randomness amplification and generation of secure bits.
[1] J. Barrett, L. Hardy, A. Kent, Phys. Rev. Lett. 95, 010503 (2005)
[2] R. Renner. R. Collbeck, Nat. Phys. 8, 450 (2012)
[3] F. G.S.L. Brandao, R. Ramanathan, A. Grudka, K. Horodecki, M. Horodecki, P. H., T. Szarek,
H. Wojewodka, Nat. Comm. 7, 11345 (2016)
[4] P. Horodecki and R. Ramanathan, Nat. Comm. 10,1701 (2019)
[5] R. Ramanathan, M. Banacki, R. Ravel Rodrigues, P. Horodecki, npj Quantum Information, 8,
119 (2022).
[6] M. Banacki, P. Mironowicz, R. Ramanathan, P. Horodecki, New J. Phys. 24, 083003 (2022)
[7] A. B. Sainz, N. Brunner, D. Cavalcanti, P. Skrzypczyk, T. Vertesi, Phys. Rev. Lett. 115, 190403 (2015)
[8] M. Banacki, R. Ramanathan, P. Horodecki, Multipartite channel assemblages,
arXiv:2205.05033 (2022).
Oct. 13, 2023
Maryanthe Malliaris :
3 p.m. in 636 SEO
Abstract
The ultraproduct gives a way of averaging an infinite sequence
of mathematical structures, such as fields, graphs, or linear orders.
The talk will be about the strength of such a construction.
Oct. 20, 2023
Kathryn Mann :
3 p.m. in 636 SEO
Abstract
Anosov flows are a fascinating class of dynamical systems, generalizing and including geodesic flows on manifolds of negative curvature. These systems exhibit "local chaos but global stability" - individual orbits diverge wildly, but the systems as a whole are stable under perturbation. This stability means there is some hope to classify them by discrete algebraic invariants. Even on 3-dimensional spaces, this is an interesting and challenging problem. In this talk, I will describe some of the history and motivation for classification (dating back to work of Anosov and Smale in the 60s), connections with low-dimensional geometric topology, and will describe recent joint work with Barthelmé, Bowden, Frankel and Fenley (in various combinations) giving answering one thread of the classification problem in dimension 3.
Oct. 27, 2023
Laura Schaposnik :
3 p.m. in 636 SEO
Abstract
In the world of science, choosing a path isn't always straightforward. The first segment of this presentation delves into the intricacies of a mathematician's career path. We explore the myriad questions, challenges, and decisions encountered when contemplating a life within the academic halls or steering towards other avenues. Transitioning to the second half, we'll dive into how geometry plays a role in different areas of science. I will share insights from my recent research endeavors, illustrating how geometric concepts offer a unique lens to comprehend phenomena, whether they span the minuscule scale of viral architectures or the expansive dynamics of molds and societal patterns.
Nov. 3, 2023
Jon Chaika :
3 p.m. in 636 SEO
Abstract
For about 2 decades the horocycle flow on strata of translation surfaces was studied, very successfully, in analogy with unipotent flows on homogeneous spaces, which by work of Ratner, Margulis, Dani and many others, have striking rigidity properties. In the past decade Eskin-Mirzakhani and Eskin-Mirzakhani-Mohammadi proved some analogous rigidity results for SL(2,R) and the full upper triangular subgroup on strata of translation surfaces. This talk will begin by introducing ergodic theory and translation surfaces and then it will describe some of the previously mentioned rigidity before moving onto its goal, that many such rigidity results fail for the horocycle flow on strata of translation surfaces. Time permitting we will also describe some rigidity result for special sub-objects in strata of translations surfaces. This will include joint work with Osama Khalil, John Smillie, Barak Weiss and Florent Ygouf.
Nov. 10, 2023
Sara Maloni :
3 p.m. in 636 SEO
Abstract
The Teichmüller space of a surface S is the space of marked hyperbolic structure on S, up to equivalence. By considering the holonomy representation of such structures, this space can also be seen as a connected component of representations from the fundamental group of S into Isom(H^2). Generalizing this point of view, Higher Teichmüller Theory studies connected components of representations from the fundamental group of S into Lie groups of rank greater than 1.
We will discuss parts of the classical theory of deformations of geometric structures, Higher Teichmüller Theory and the notion of Anosov representation. We will conclude by describing the fact that these representations correspond to deformation of certain geometric structures, where we will also discuss a recent joint work with Alessandrini, Tholozan and Wienhard.
Dec. 5, 2023
Lior Gishboliner :
3 p.m. in 636 SEO
Abstract
The key question of study in extremal graph theory can be stated as follows: How dense should a graph (or hypergraph) be globally in order to guarantee that it contains a given local structure? This global-to-local theme is also a key feature of the field of property testing, which deals with the design of randomized "election-polling"-type algorithms which infer global information about a graph from local samples. I will survey several interconnected topics in extremal graph theory and property testing, focusing on the influential Brown-Erdos-Sos conjecture and the removal lemma. I will also present new results on these topics.
Jan. 10, 2024
Siting Liu :
3 p.m. in 636 SEO
Abstract
This presentation explores optimization strategies for improving both partial differential equation (PDE) computations and score-based generative models (SGM). In the realm of numerical computations, we introduce a saddle point framework that capitalizes on the inherent structure of PDEs. Integrated seamlessly with existing discretization schemes, this framework eliminates the necessity for nonlinear inversions, paving the way for efficient parallelization. Shifting our focus to SGM, we delve into the mathematical foundations of the Wasserstein proximal operator (WPO). Specifically, we express it as the Wasserstein proximal operator of cross-entropy. By leveraging the PDE formulation of WPO, we propose a WPO-informed score model that demonstrates accelerated training and reduced data requirements.
Jan. 11, 2024
Dmitriy Kunisky :
3 p.m. in 636 SEO
Abstract
I will present a line of work on the computational complexity of several algorithmic tasks on random inputs, including hypothesis testing, sampling, and "certification" for optimization problems (where an algorithm must output a bound on a problem's optimum rather than just a high-quality solution). Surprisingly, these diverse tasks admit a unified analysis involving the same two main ingredients. The first is the study of algorithms that output low-degree polynomial functions of their inputs. Such algorithms are believed to be optimal for many statistical tasks and can be understood with the theory of orthogonal polynomials, leading to strong evidence for the hardness of certain hypothesis testing problems. The second is a strategy of "planting" unusual structures in problem instances, which gives reductions from hypothesis testing to tasks like sampling and certification. I will focus on examples of the latter motivated by statistical physics: (1) sampling from Ising models, and (2) certifying bounds on the Hamiltonian of the Sherrington-Kirkpatrick spin glass model.
Next, by examining the sum-of-squares hierarchy of semidefinite programs, I will demonstrate how reasoning with planted solutions can show computational hardness of certification problems not only in random settings under strong distributional assumptions, but also for more generic problem instances. As an extreme example, I will show how some of the above ideas may be completely derandomized and applied in a deterministic setting. Using as a testbed the long-standing open problem in number theory and Ramsey theory of bounding the clique number of the Paley graph, I will give an analysis of semidefinite programming that suggests both new theoretical approaches to proving stronger bounds on the clique number and refined notions of pseudorandomness capturing deterministic versions of phenomena from random matrix theory.
Jan. 12, 2024
Tomer Galanti :
3 p.m. in 636 SEO
Abstract
In this talk, we delve into several fundamental questions in deep learning. We start by addressing the question, "What are good representations of data?" Recent studies have shown that the representations learned by a single classifier over multiple classes can be easily adapted to new classes with very few samples. We offer a compelling explanation for this behavior by drawing a relationship between transferability and an emergent property known as neural collapse. Additionally, we explore why certain architectures, such as convolutional networks, outperform fully-connected networks, providing theoretical support for how their inherent sparsity aids learning with fewer samples. Lastly, I present recent findings on how training hyperparameters implicitly control the ranks of weight matrices, consequently affecting the model's compressibility and the dimensionality of the learned features.
Additionally, I will describe how this research integrates into a broader research program where I aim to develop realistic models of contemporary learning settings to guide practices in deep learning and artificial intelligence. Utilizing both theory and experiments, I study fundamental questions in the field of deep learning, including why certain architectural choices improve performance or convergence rates, when transfer learning and self-supervised learning work, and what kinds of data representations are learned with Stochastic Gradient Descent.
Jan. 19, 2024
Anton Bernshteyn :
3 p.m. in 636 SEO
Abstract
In this talk I will discuss a recently discovered connection between two seemingly very disparate fields: distributed computing and descriptive combinatorics. Distributed computing is the area of computer science concerned with problems that can be solved efficiently by a (finite) decentralized network of processors. Descriptive combinatorics, on the other hand, studies combinatorial problems on infinite graphs under additional topological or measure-theoretic regularity constraints and is largely motivated by questions in set theory, ergodic theory, and topological dynamics. It turns out that these two areas are intimately related to each other, and there are formal ways of translating results from one context to the other one. I will survey what is known about this connection and outline some open problems.
Jan. 22, 2024
Emmanouil-Vasileios Vlatakis Gkaragkounis :
3 p.m. in 636 SEO
Abstract
Traditional computing sciences have made significant strides using tools like Complexity and Worst-Case Analysis over the past decades. However, the rise of Machine Learning has brought a renewed focus on complex optimization problems found in diverse areas such as RoboSoccer, image generation, autonomous vehicles, and multi-objective logistics optimization. While theoretically challenging, these problems often prove more manageable in real-world scenarios, thanks to modern Machine Learning techniques which surprisingly often rely on straightforward methods like Local Search and Gradient Descent.
In this talk, I'll explore why these seemingly simple algorithms are effective in complex environments. We'll look into developing a theory that moves beyond traditional analysis, connecting theoretical concepts with practical applications. The discussion will also cover decision-making in scenarios with conflicting incentives and uncertainty, using advanced methods from Optimization, Statistics, and Game Theory. We'll delve into the dynamics of strategic decision-making where data uncertainty and opposing incentives play a crucial role.
Feb. 2, 2024
Anna Mazzucato :
3 p.m. in 636 SEO
Abstract
Stirring and mixing in fluids, specifically incompressible fluids, have important consequences on many physical and biological processes, from dispersal of pollutants to transport of nutrients. From a mathematical point of view, mixing can be studied in different contexts, from ergodic theory to homogenization.
In this talk, I will present a quantitative approach to mixing that arises in the analysis of partial differential equations. In this context, mixing is related to irregular transport by non-Lipschitz vector fields and, when combined with diffusion, it may lead to enhanced dissipation. A variety of techniques have been employed in the literature to study these mechanisms, from geometric analysis to optimal transport to spectral theory and probability.
I will first discuss examples of incompressible flows that mix optimally in time. Then, I will show how these examples lead to loss of regularity for solutions of transport equations. Lastly, I will discuss enhanced dissipation and examples of flows that lead to enhanced dissipation for advection-(hyper)diffusion equations using resolvent estimates.
Feb. 9, 2024
Gabriel Conant :
2 p.m. in 636 SEO
Abstract
In 2011, Malliaris and Shelah proved a strong form of Szemeredi's regularity lemma for the class of "stable graphs", which are graphs omitting a certain special subgraph called a ``half-graph". A group theoretic analogue of their result for finite abelian groups was later obtained by Terry and Wolf using Fourier analytic methods from additive combinatorics. A suitable generalization to arbitrary finite groups was then proved by myself, Pillay, and Terry using model theoretic methods. This talk will focus on an analytic analogue of stability defined for functions, rather than graphs. Roughly speaking, the main result of the talk says that if G is amenable, then any stable function on G is almost constant on all translates of a unitary Bohr set in G of bounded complexity. The proof of this result uses ingredients from topological dynamics and continuous model theory. I will also explain how this result leads to a short proof of Bogolyubov’s Lemma for arbitrary amenable groups. This is joint work with Anand Pillay.
Chantal David :
3 p.m. in 636 SEO
Abstract
Gauss sums are fundamental objects in number theory. Quadratic Gauss sums
were studied by Gauss, and after many attempts, Gauss gave a simple
formula depending only on the argument of the Gauss sums modulo 4. Higher
degree Gauss sums seem to behave differently. Based on numerical
evidence, it was suggested by Kummer (1846) that the angles of cubic
Gauss at prime arguments are not equidistributed, and exhibit a bias
towards positive values. More extensive numerical testings seemed to
indicate that the bias does not persist, and that cubic Gauss sums are
indeed equidistributed, which was proven by Heath-Brown and Patterson
(1979). We will explain in this talk what is involved in proving
equidistribution of cubic Gauss sums, in particular why it took more
than 130 years after Kummer’s observations.
The talk will be accessible to a general mathematical audience, including
graduate students.
Feb. 16, 2024
Mark Rudelson :
3 p.m. in 636 SEO
Abstract
The existence and the number of solutions of a system of polynomial equations in n variables over an algebraically closed field is a classical topic in algebraic geometry. Much less is known about the existence of solutions of a system of polynomial equations over reals. Any such problem can be reduced to a system of quadratic equations by introducing auxiliary variables. Due to the generality of the problem, a computationally efficient algorithm for determining whether a real solution of a system of quadratic equations exists is believed to be impossible. We will discuss a simple sufficient condition for the existence of a solution which can be efficiently checked. While the problem and the condition are of algebraic nature, the approach lies entirely within the analysis/probability realm and relies on tools from Fourier analysis and concentration of measure.
Joint work with Alexander Barvinok.
Feb. 23, 2024
Carmen Rovi :
3 p.m. in 636 SEO
Abstract
In this talk, we will be concerned with a relation between TQFTs and the controlled cut-and-paste invariants of manifolds introduced by Karras, Kreck, Neumann, and Ossa. The controlled cut-and-paste invariants (SKK invariants) are functions on the set of smooth manifolds whose values on cut-and-paste equivalent manifolds differ by an error term depending only on the gluing diffeomorphisms. I will present a natural group homomorphism between the group of invertible TQFTs and the group of SKK invariants and describe how these groups fit into a split exact sequence. We conclude in particular that all positive real-valued SKK invariants can be realized as restrictions of invertible TQFTs.
March 1, 2024
Adam Topaz :
3 p.m. in 636 SEO
Abstract
Recent years have witnessed a significant increase in efforts to
digitize and formalize mathematics. Various motivations underlie this
newfound interest, including the formal verification of complex
mathematical proofs, the construction of cohesive libraries of
mathematical structures and theorems, and the anticipated promise of
interaction with AI. This paradigm shift has enabled mathematicians
working within these formal systems to approach the discipline in new
ways. For many, the formalization of mathematics has even redefined
what they find mathematically meaningful and interesting. This talk
will highlight several key aspects of this new approach to doing
mathematics.
March 8, 2024
Izzet Coskun :
3 p.m. in 636 SEO
Abstract
The configuration space of n points in the plane is the space of n-unordered tuples of distinct points. Grothendieck's Hilbert scheme provides a smooth compactification of this space. In this talk, I will focus on the question: What is the most special codimension one position that n points can lie in? For example, three points are typically not collinear, but in codimension one they can be collinear. This simple question will lead us to a tour of some fun mathematics ranging from moduli spaces of stable sheaves on the plane to fractal curves and palindromic numbers. This talk is based on joint work with Jack Huizenga and Matthew Woolf.
March 15, 2024
Justin Sawon :
3 p.m. in 636 SEO
Abstract
Given an elliptic curve E defined over Q, its Tate-Shafarevich group classifies torsors over E. These are genus 1 curves that have points `locally' (p-adically) but not `globally' (rational points). The same ideas apply in complex geometry if we replace the field Q by the function field of a complex manifold, e.g., CP^1. Then E become an elliptic surface fibred over CP^1 which admits a section, and its torsors are elliptic surfaces that are isomorphic to E locally over the base, but not globally.
In this talk we explain how Tate-Shafarevich twists can be applied in holomorphic symplectic geometry. Given a `Lagrangian fibration', i.e., a fibration X->B whose general fibres are Lagrangian abelian varieties, Tate-Shafarevich twists can produce new examples of holomorphic symplectic manifolds, and also reveal unexpected relations between these spaces.
March 29, 2024
Cosmin Pohoata :
3 p.m. in 636 SEO
Abstract
Given an integer $n \geq 3$, the Heilbronn triangle problem asks for the smallest number $\Delta = \Delta(n)$ such that in every configuration of $n$ points in the unit square $[0,1]^2$ one can always find three among them which form a triangle of area at most $\Delta$. A trivial upper bound of the form $\Delta = O(1/n)$ follows from the simple observation that if one partitions $[0,1]^2$ into $n/2$ vertical strips of width $2/n$, then one of these strips must contain at least $3$ of the points. The problem of improving upon this basic observation is a difficult one, with a long and rich history. We will survey some of this history and discuss with some new connections between this problem and various themes in incidence geometry, harmonic analysis, and projection theory. Based on joint work with Alex Cohen and Dmitrii Zakharov.
April 5, 2024
Marcus Michelen, James Freitag, Nicole Looper, and Dhruv Mubayi (UIC) :
3 p.m. in 636 SEO
Abstract
Applying for postdoc positions can often be a daunting task. In a short presentation, Marcus Michelen will give a 10-15 minute overview of the basic logistics of applying for postdocs in the US: it will cover what materials and reference letters are needed; what the timeline is like; and some basic tips. The presentation will be followed by a question-and-answer panel with James Freitag, Nicole Looper and Dhruv Mubayi.
April 12, 2024
Thomas Hou :
3 p.m. in 636 SEO
Abstract
Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this talk, I will present some numerical evidence that the 3D Navier-Stokes equations develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of $10^7$. This potentially singular behavior is induced by a potential finite time singularity of the 3D Euler equations. Unlike the Hou-Luo blowup scenario, the potential singularity of the 3D Euler and Navier-Stokes equations occurs at the origin. We have applied several blowup criteria to study the potentially singular behavior of the Navier-Stokes equations. The Beale-Kato-Majda blow-up criterion, the blowup criteria based on the growth of enstrophy and negative pressure, the Ladyzhenskaya-Prodi-Serrin regularity criteria all seem to imply that the Navier-Stokes equations develop potentially singular behavior. Finally, we present some new numerical evidence that a variant of the axisymmetric Navier-Stokes equations with time dependent fractional dimension develops nearly self-similar blowup with maximum vorticity increased by a factor of $10^{32}$.
April 19, 2024
Patrick Lutz :
3 p.m. in 636 SEO
Abstract
The field of computability theory studies the complexity of uncomputable problems. In this study, a special role is played by the Halting Problem. Not only is it the first problem proved to be uncomputable, it also seems to be the simplest "natural" uncomputable problem. Martin's Conjecture is a long-standing open problem in computability theory which gives a partial explanation for this phenomenon. A key idea behind Martin's Conjecture is to view the Halting Problem not just as a problem, but as an operator on problems, which takes any problem to a strictly harder one. Martin's Conjecture consists of a classification of such operators, which says, in part, that the Halting Problem is the minimal non-trivial operator. I will discuss the background and motivation for Martin's Conjecture, as well as recent progress by Benjamin Siskind and myself which essentially completes a proof of the conjecture for a special class of operators called "order-preserving" (i.e. those which preserve the relative complexity of problems).
April 26, 2024
Ruxandra Moraru :
3 p.m. in 636 SEO
Abstract
A holomorphic symplectic manifold is a complex manifold $X$ together with a closed, non-degenerate holomorphic 2-form $\Omega$. The top power of $\Omega$ gives a trivialisation of the canonical bundle so that $X$ has trivial first Chern class. In the context of K\”ahler geometry, such manifolds play a very important role due to the Bogomolov covering theorem, which states that any compact Kahler manifold with vanishing first Chern class has a covering that splits as the product of Calabi–Yau manifolds, complex tori and irreducible holomorphic symplectic manifolds. Among these, the last two are, in fact, compact holomorphic symplectic manifolds. Furthermore, irreducible holomorphic symplectic manifolds correspond to compact hyperk\”ahler manifolds in the K\”ahler setting. In general, finding compact holomorphic symplectic manifolds is very difficult. In this talk, I will present some classical constructions of holomorphic symplectic manifolds in various geometric contexts and discuss some recent developments in finding compact examples in the non-Kahler setting.
Sept. 6, 2024
Emily Barnard :
3 p.m. in 636 SEO
Abstract
The pop-stack sorting method takes an ordered list or permutation and reverses each descending run without changing their relative positions. In this talk we will review recent combinatorial results on the pop-stack sorting method, and we will extend the pop-stack sorting method to certain pattern avoiding permutations, called c-sortable. This talk will be accessible to all.
Sept. 13, 2024
Teena Gerhardt :
3 p.m. in 636 SEO
Abstract
The field of algebraic topology has exposed deep connections between topology and algebra. One example of such a connection comes from algebraic K-theory. Algebraic K-theory is an invariant of rings, defined using tools from topology, that has important applications to algebraic geometry, number theory, and geometric topology. Algebraic K-groups are difficult to compute, but advances in algebraic topology have led to many recent computations which were previously intractable. In this talk I will introduce algebraic K-theory and its applications, and discuss recent advances in this field.
Sept. 20, 2024
Maria Chudnovsky :
3 p.m. in 636 SEO
Abstract
Tree decompositions are a powerful tool in both structural
graph theory and graph algorithms. Many hard problems become tractable if
the input graph is known to have a tree decomposition of bounded
“width”. Exhibiting a particular kind of a tree decomposition is also
a useful way to describe the structure of a graph.
Tree decompositions have traditionally been used in the context of forbidden
graph minors; studying them in connection with graph containment relations of
more local flavor (such as induced subgraph or induced minors) is a relatively
new research direction. In this talk we will discuss recent progress in this
area, touching on both the classical notion of bounded tree-width, and
concepts of more structural nature.
Sept. 27, 2024
Jeffrey Danciger :
3 p.m. in 636 SEO
Abstract
[The speaker will be part of the Big ideas in Dynamics and Geometry workshop.]
The Auslander Conjecture is an analogue of Bieberbach’s theory of Euclidean crystallographic groups in the setting of affine geometry. It predicts that a complete affine manifold (a manifold equipped with a complete torsion-free flat affine connection) which is compact must have virtually solvable fundamental group. The conjecture is known up to dimension six, but is known to fail if the compactness assumption is removed, even in low dimensions. We discuss some history of this conjecture, give some basic examples, and then survey some recent advances in the study of non-compact complete affine manifolds with non-solvable fundamental group.
Tools from the deformation theory of pseudo-Riemannian hyperbolic manifolds and also from higher Teichm\"uller theory will enter the picture.
Oct. 11, 2024
Maya Banks and Félix Lequen :
3 p.m. in 636 SEO
Abstract
2nd speaker: Felix Lequen
Title: Bourgain's construction of finitely supported measures with regular Furstenberg measure
Abstract: In this talk, I will explain how to study the regularity of a very natural object in random dynamics called the Furstenberg measure. I will first try to explain how this problem relates to natural fractal-like structures we find in dynamics in general, and why it is interesting and very rich in its own right, using the case of Bernoulli convolutions. These can be seen as measure generalizations of the usual middle-third Cantor set, in particular when there are overlaps. In this context Erdős observed in the 30s that there are some non-trivial phenomena, for instance with a number-theoretic flavour.
I will then move on to the random iterations of matrices, that is a random walk on matrices, focusing on the case of SL(2, R), where the Furstenberg measure alluded to above contains much of the interesting asymptotic information. It had been conjectured that for a finitely supported random walk on matrices, the Furstenberg measure is never absolutely continuous with respect to the Lebesgue measure. Following a construction of Bourgain, I will show how to construct examples where the Furstenberg measure is absolutely continuous and its density has high regularity. This construction relies essentially on a spectral gap property due to Boutonnet-Ioanna-Salehi--Golsefidy that I will introduce.
Oct. 18, 2024
Daniel Berwick Evans :
3 p.m. in 636 SEO
Abstract
Modular forms appear in a wide variety of contexts in physics and mathematics. For example, they arise in two dimensional quantum field theories as certain expectation values. In algebraic topology, they emerge in the study of elliptic cohomology theories. A long-standing conjecture suggests that these two appearances of modular forms are intimately related. This talk will provide an introduction to these ideas.
Oct. 25, 2024
Christian Wolf :
3 p.m. in 636 SEO
Abstract
The computability of dynamically defined objects has been a subject of
intensive study over the past two decades. This includes important results
concerning the computability of invariant sets (e.g. Julia sets), entropies,
natural invariant measures, Lyapunov exponents, etc. In this talk we consider
symbolic dynamical systems given by the shift map on a finite alphabet-shift
space $X$. We identify the computability of the topological entropy
$h_{\rm top}(X)$ and topological pressure $P_{\rm top}(X,\phi)$ as a function
of the shift space $X$ and potential $\phi$. This question has previously
been studied by Burr et al., Hertling and Spandl, and Spandl in some
special cases. In this talk we address the computability question for
general shift spaces. One of our results states that the entropy function
$X\mapsto h_{\rm top}(X)$ is computable at a shift space $X_0$ iff $X_0$
has zero topological entropy.
Nov. 1, 2024
Amanda Young :
3 p.m. in 636 SEO
Abstract
Materials exhibiting quantum mechanical effects hold the promise of developing revolutionary new technologies. A vital component to this progress is understanding the phase of individual quantum systems, which groups together physical systems that exhibit similar properties and characteristics. One of the fundamental quantities in this classification is whether or not there is a spectral gap above the ground state energy of the system. In this talk, we will discuss how these phases are modeled, how the gap is defined and its role in phase classification, and highlight some of the key past and current research related to this vital task.
Nov. 8, 2024
Juanita Pinzon Caicedo :
3 p.m. in 636 SEO
Abstract
The fundamental group is one of the most powerful invariants to distinguish closed three-manifolds, and the existence of non-trivial homomorphisms $\pi_1(M)\to SU(2)$ is a great way of measuring the non-triviality of a three-manifold $M$. It is known that if an integer homology 3-sphere is either Seifert fibered or toroidal, then irreducible representations do exist. In contrast, the existence of SU(2)-representations for hyperbolic homology spheres has not been completely established. With this as motivation, I will talk about partial progress made in the case of hyperbolic homology spheres realized as branched covers. This is joint work with Sudipta Ghosh.
Nov. 15, 2024
Noah Giansiracusa :
3 p.m. in 636 SEO
Abstract
During the pandemic I felt I needed a break from my usual research program so on a whim decided to try writing a book aimed at the general public, knowing nothing about how to do so. Unexpectedly, this launched a winding professional journey that over the past five years has included meetings with congressional staffers, guest appearances on CNN, a handful of newspaper op-eds, signing a literary agent, a friendship with a Nobel prize-winning economist, and plenty of Twitter feuds along the way. In this talk I'll share what I've learned from this adventure (including the missteps), distilling practical advice for those wishing to explore some of these professional avenues or others like them. I'll emphasize what role a mathematical background has played in a range of activities (like writing and networking) that don't outwardly appear to involve any mathematics.
Jan. 24, 2025
Chris Leininger :
3 p.m. in 636 SEO
Abstract
I will discuss joint work with Autumn Kent in which we establish the existence of purely pseudo-Anosov surface subgroups of mapping class groups. We do this by constructing a type-preserving representation of the figure eight knot group into the mapping class group of the thrice-punctured sphere. As a corollary we obtain the first examples of closed atoroidal surface bundles over surfaces.
Feb. 7, 2025
Simion Filip :
3 p.m. in 636 SEO
Abstract
For a general dynamical system, it is impossible to analyze
the behavior of all points and instead one has a description of orbits
that are typical with respect to some invariant measure; the description
of all invariant measures is equally unwieldy. For certain flows on
homogeneous spaces associated to Lie groups, the measure and topological
rigidity results initiated by Margulis and Ratner showed that it is
possible to give a useful description for the orbit of every point,
and describe all invariant measures. I will discuss some results which,
under appropriate conditions, give similar measure and topological
rigidity properties for smooth dynamical systems on general manifolds.
Joint work with Aaron Brown, Alex Eskin, and Federico Rodriguez--Hertz,
as well as David Fisher and Ben Lowe.
Feb. 21, 2025
Daniel Litt :
3 p.m. in 636 SEO
Abstract
When does an algebraic differential equation admit an algebraic solution?
This has been an animating question behind much mathematics since the
latter half of the 19th century. It is now understood (due to work of
Siegel, Grothendieck, Katz, and others) to be closely related to number
theory. I'll survey some of the conjectures and results on this topic,
and explain some recent progress in understanding what happens in the
case of non-linear algebraic differential equations--for example, the
Painlevé VI equation and Schlesinger system--in joint work with Josh Lam.
Feb. 28, 2025
Mathilde Gerbelli-Gauthier :
3 p.m. in 636 SEO
Abstract
In 2016, Viazovska proved that the $E_8$ lattice provides the optimal
sphere-packing in dimension 8, and soon after,
Cohn--Kumar--Miller--Radchenko--Viazovska proved the analogous result
for the Leech lattice in dimension 24. Viazovska's breakthrough came
through the solution of a Fourier interpolation problem: she constructed
a function $f$ such that $f$, its Fourier transform, and their first
derivatives take on specific values at square roots of natural numbers.
Prior to this, Radchenko and Viazovska had solved a "toy case"—a
Fourier interpolation result for even Schwartz functions on the
real line. In this talk, I will explain how this one-dimensional
version can be understood through the lens of the Segal--Shale--Weil
representation, an infinite-dimensional representation that
originally arose in the context of quantum mechanics.
March 14, 2025
Ji Zhu :
3 p.m. in 636 SEO
Abstract
Most statistical models for networks focus on pairwise interactions between nodes. However, many real-world networks involve higher-order interactions among multiple nodes, such as co-authors collaborating on a paper. Hypergraphs provide a natural representation for these networks, with each hyperedge representing a set of nodes. The majority of existing hypergraph models assume uniform hyperedges (i.e., edges of the same size) or rely on diversity among nodes. In this work, we propose a new hypergraph model based on non-symmetric determinantal point processes. The proposed model naturally accommodates non-uniform hyperedges, has tractable probability mass functions, and accounts for both node similarity and diversity in hyperedges. For model estimation, we maximize the likelihood function under constraints using a computationally efficient projected adaptive gradient descent algorithm. We establish the consistency and asymptotic normality of the estimator. Simulation studies confirm the efficacy of the proposed model, and its utility is further demonstrated through edge predictions on several real-world datasets.
April 4, 2025
Ben Davison :
3 p.m. in 636 SEO
Abstract
BPS cohomology is a cohomology theory that "categorifies" refined BPS invariants associated to 3-Calabi-Yau categories, in the sense that this cohomology recovers these invariants after passing to the (virtual) Poincaré polynomial of the BPS cohomology. As well as categorifying refined BPS state counts, this cohomology turns out to have a rich algebraic structure, with links to various constructions and central objects in quantum algebra, cluster algebras and geometric representation theory. I will survey the construction of BPS cohomology, as well as applications in the above areas and beyond.
April 11, 2025
Marcus Michelen, Daniel Groves, Matthew Harrison-Trainor, Amelia Pompilio :
3 p.m. in 636 SEO
Abstract
Applying for postdoc positions can often be a daunting task. In a short presentation, Marcus Michelen will give a 10 minute overview of the basic logistics of applying for postdocs in the US: it will cover what materials and reference letters are needed; what the timeline is like; and some basic tips. The presentation will be followed by a question-and-answer panel with Daniel Groves, Matthew Harrison-Trainor, and Amelia Pompilio
April 18, 2025
Alexander Bertoloni Meli :
3 p.m. in 636 SEO
Abstract
I will introduce and motivate the local Langlands program for
p-adic groups starting from the perspective of understanding the
absolute Galois group of the p-adic numbers. I'll explain why we
might want to upgrade the Langlands correspondence to an
equivalence of categories as conjectured by Fargues and Scholze
and discuss what progress has been made in proving this conjecture.
May 2, 2025
Tsachik Gelander :
3 p.m. in 636 SEO
Abstract
A mixed identity in group G is an equation W(x)=1 where W is a non-trivial word in the free product G∗⟨x⟩, which is satisfied for all x∈G. Mixed Identity Free (MIF) means that no such identity holds on G. When G has no mixed identities, one wishes to find such x effectively (w.r.t. the word metric). Set
f(n)=min { | g | : g∈G , W(g)≠1 for all W∈B(n) }
where B(n) is the n-th ball in G∗⟨x⟩.
If f is sub-exponential, there are outstanding applications for the reduced C*-algebra of the group, especially when the group also has rapid decay.
Recently, Elayavalli and Schafhauser gave a negative answer for the C*-algebraic Tarski problem by studying this property for free groups. More recently, Itamar Vigdorovich extended their work to uniform lattices in SL(n,R). What we proved is:
Theorem 1. For a f.g. linear group \Gamma with MIF, the function f is linear (i.e. f(n)<Cn).
If the Zariski closure G is a classical group, then \Gamma is MIF, provided G is PSL(n), or G=SP_{2r} and \Gamma has no elements of order 2, or G=SO(n) and \Gamma has no elements g for which g+g^{-1} is a scalar. Along the way, we proved a new variant of the super approximation theorem, which is of independent interest.
This is a joint work with Nir Avni
Sept. 5, 2025
Marina Logares :
3 p.m. in 636 SEO
Abstract
What do water waves have to do with mirror symmetry, or even with Kovalevskaya tops? All of them are, or are closely related to, integrable systems, in particular to a class of algebraically completely integrable systems known as Hitchin systems. In this talk, I will explore how Hitchin systems provide a unifying framework connecting seemingly disparate areas of mathematics and physics. The presentation will be accessible to a general mathematical audience, while offering concrete connections for specialists in geometry, topology, and algebraic geometry.
Sept. 26, 2025
Daniel Groves :
3 p.m. in 636 SEO
Abstract
Due to a vast amount of work over the last decades, the fundamental groups of
3-manifolds are by now very well understood. I will focus on the following
(wide open) question: When is a discrete group the fundamental group of a
compact 3-manifold? I'll discuss the background to this question, what is
known in various dimensions, and then focus on the case of greatest interest
in 3 dimensional topology - the hyperbolic case. Finally, I'll report on
some recent work around this question in joint work with Haissinsky,
Manning, Osajda, Sisto, and Walsh.
Oct. 17, 2025
Artem Chernikov :
3 p.m. in 636 SEO
Abstract
Finite VC-dimension, a combinatorial property of families of sets, was discovered simultaneously in the 70's by Vapnik and Chervonenkis in probabilistic learning theory, and by Shelah in model theory (where it is called NIP). It plays an important role in several areas including machine learning, combinatorics, mathematical logic, functional analysis and topological dynamics. A higher arity generalization of VC-dimension for families of sets in n-fold product spaces (i.e. a bound on the sizes of n-dimensional boxes that can be shattered) is implicit in Shelah's work on n-dependent theories in model theory. Following some preliminary work in Chernikov, Palacin, Takeuchi '14, in Chernikov, Towsner '20 we developed aspects of higher-arity VC-theory, including a generalization of Haussler's packing lemma for families of sets (and real-valued functions) of bounded VC_n-dimension. Probably Approximately Correct (PAC) learning is a classical framework for mathematical analysis of machine learning, and PAC learnability is famously characterized by finite VC dimension. Generalizing this, we demonstrate that finite VC_n dimension characterizes higher arity PAC learning (PAC_n learning) in n-fold product spaces with respect to product measures introduced by Kobayashi, Kuriyama and Takeuchi '15. Joint work with Henry Towsner.
Nov. 7, 2025
Greg Baldi :
3 p.m. in 636 SEO
Abstract
Many geometric spaces come equipped with a natural collection of special
submanifolds that reflect their internal symmetries. Examples include
abelian varieties with their sub-abelian varieties, locally symmetric
spaces with totally geodesic subspaces, period domains with sub–period
domains, and strata of abelian differentials with affine invariant
submanifolds.
In recent years, significant progress has been made in understanding
such structures through the lens of unlikely intersections and
functional transcendence. I will outline the general framework of
variations of Hodge structures and period domains, and explain how the
so-called completed Zilber–Pink philosophy provides a unifying way to
describe the qualitative behaviour of these special loci. This
perspective reveals deep connections between arithmetic geometry, Hodge
theory, and dynamical systems.
Nov. 14, 2025
Nir Avni :
3 p.m. in 636 SEO
Abstract
I will talk about two new rigidity phenomena for arithmetic groups.
The first, model theoretic, is about axiomatizing such a group using
first order logic. The second is about conjugation-invariant norms on the group.
Both rigidity results rely on studying the set-theoretic products of conjugacy
classes. Based on joint works with Alex Lubotzky and Chen Meiri.
Dec. 5, 2025
Lior Gishboliner :
3 p.m. in 636 SEO
Abstract
Regularity and VC-dimension are two fundamental notions with many
applications in combinatorics and beyond. These notions are related via
the result that graphs of bounded VC-dimension have (small) partitions
where most pairs of parts have density close to 0 or 1. Recent work has
generalized this to hypergraphs, but the quantitative aspects of these
results are still far from fully understood. I will present some new
results on this problem. Joint work with Asaf Shapira and Yuval Wigderson.
Feb. 6, 2026
Haotian Jiang :
3 p.m. in 636 SEO
Abstract
The Beck-Fiala Conjecture asserts that any set system of n elements with degree k has combinatorial discrepancy $O(\sqrt{k})$. A substantial generalization is the Komlós Conjecture, which states that any m by n matrix with columns of unit Euclidean length has discrepancy O(1).
In this talk, we describe an $\tilde{O}(\log^{1/4} n)$ bound for the Komlós problem, improving upon the $O(\log^{1/2} n)$ bound due to Banaszczyk from 1998. We will also see how these ideas can be used to resolve the Beck-Fiala Conjecture for $k \geq \log^2 n$, and give a $\tilde{O}(k^{1/2} + \log^{1/2} n)$ bound for smaller k, which improves upon Banaszczyk's $O(k^{1/2} \log^{1/2} n)$ bound. These results are based on a new technique of "Decoupling via Affine Spectral Independence" in designing rounding algorithms, which might also be useful in other contexts.
This talk is based on joint work with Nikhil Bansal (University of Michigan).
Feb. 20, 2026
Wes Pegden :
3 p.m. in 636 SEO
Abstract
What can we understand about probability spaces on "nice" partitions of a
geometric region? Can we design efficient samplers? Can we at least detect
extreme outliers? These questions have become particularly salient in the
past several years as the techniques developed by mathematicians are now
applied to conduct statistical analyses of things like U.S. political
districtings. We will discuss some recent developments on probability
spaces defined by geometric constraints, including positive and negative
results on the mixing times of relevant Markov chains, Markov chain methods
which eschew mixing-time requirements, and direct sampling methods.
March 20, 2026
Candice Price :
3 p.m. in Lecture Center D1
Abstract
Mathematical modeling is an effective resource for biologists since it provides ways to simplify, study, and understand the complex systems common in biology and biochemistry. Many mathematical tools can be applied to biological problems, some traditional and some more novel, all innovative. This presentation will review the mathematical tools I use to model and study biological issues related to DNA-protein interactions.
April 10, 2026
Wei Zheng :
3 p.m. in 636 SEO
Abstract
The Shapley value, a fundamental concept in cooperative game theory, provides a fair allocation of cooperative gains or costs among players. However, computing Shapley values for a game with $d$ players requires evaluating all $2^d$ coalitions, which is computationally infeasible for large $d$. This difficulty is exacerbated in modern applications such as artificial intelligence, data science, and genomics, where evaluating the value of even a single coalition can be costly. To enable fast approximation and probabilistic inference of the Shapley value, we propose the Bayesian framework, where the Gaussian process is adopted to infer unobserved coalition values. The posterior distribution of the coalition values are then transformed into that of the Shapley values, allowing both point estimation and uncertainty quantification. We derive theoretical results showing that the computational complexity of posterior evaluation can be reduced from exponential to polynomial order. To further improve efficiency, we integrate experimental design principles to select coalitions that minimize posterior variances. Compared with existing approaches, the proposed method offers three main advantages: (i) support for statistical inference, (ii) accurate estimation of Shapley values using as few as $d^2-d+1$ coalition evaluations, and (iii) robustness across a wide range of cooperative games. Simulation studies and case analyses demonstrate that the proposed approach achieves higher accuracy and efficiency than existing methods under comparable evaluation cost.
April 17, 2026
Dr. Yufeng Liu :
3 p.m. in 636 SEO
Abstract
Modern decision-making systems, from online marketplaces to large language models (LLMs), increasingly rely on high-dimensional human feedback, where heterogeneous user preferences and massive feature spaces pose major challenges for statistical efficiency and alignment. In this talk, I will present low-rank reinforcement learning (RL) methods that exploit latent structures in human feedback to enable scalable and theoretically grounded learning. In the first part, we study the dynamic assortment problem in high-dimensional e-commerce and show how a low-rank structure in user–item interactions reduces the complexity of estimating personalized utilities and enables efficient exploration–exploitation strategies with provable regret guarantees. In the second part, we extend these ideas to reinforcement learning from human feedback (RLHF) in large-scale contextual environments, proposing a low-rank contextual framework that accommodates diverse user preferences and complex latent spaces in LLMs while providing theoretical guarantees on sample efficiency and robustness under distribution shifts.
April 24, 2026
Gabe Conant, Matthew Harrison-Trainor, Dhruv Mubayi, and Jagerynn Verano :
3 p.m. in 636 SEO
Abstract
Applying for postdoc positions can often be a daunting task. Matthew Harrison-Trainor will give a short presentation giving an overview of the basics of applying for postdocs: it will cover what materials and reference letters are needed; what the timeline is like; and some basic tips. The presentation will be followed by a question-and-answer panel with panelists Gabe Conant, Dhruv Mubayi, and Jagerynn Verano.