# Colloquia, AY 2013-2014

Thursdays 4:10 pm in 5211 Stevenson Center, unless otherwise noted
Tea at 3:30 pm in 1425 Stevenson Center

September 5, 2013

Welcome event, no colloquium

September 12, 2013

Mark Rudelson (University of Michigan)

Non-asymptotic approach in random matrix theory

Random matrix theory studies the asymptotics of the spectral distributions of families of random matrices, as the sizes of the matrices tend to infinity. Derivation of such asymptotics frequently requires analyzing the spectral properties of random matrices of a large fixed size, especially of their singular values.
We will discuss several recent results in this area concerning matrices with independent entries, as well as random unitary and orthogonal perturbations of a fixed matrix. We will also show an application of the non-asymptotic random matrix theory to estimating the permanent of a deterministic matrix.

Contact person:
Alex Powell

September 26, 2013

Olga Kharlampovich (Hunter College, CUNY)

A survey of algebraic geometry and model theory for free and hyperbolic groups.

I will survey results of Kharlampovich–Miasnikov and Sela on first-order theories of free and hyperbolic groups. I will show that in the presence of “negative curvature” in groups, there exists a robust algebraic geometry and the principal Tarski-type problems are decidable. In particular, there is an algorithm for the elimination of quantifiers (to boolean combinations of AE-formulas). I will also give a description of definable sets in free and hyperbolic groups (joint result with Miasnikov). This solves Malcev’s problem from 1965.

Contact person:
Denis Osin

October 3, 2013

Stavros Garoufalidis (Georgia Tech)

The stable coefficients of the Jones polynomial of a link

The Jones polynomial of a link is a finite collection of integers placed at different degrees. We propose a structure theorem for the (stable) coefficients of an alternating link in terms of a flag algebra of graphs, verify it for the first 4 coefficients and present further experimental evidence for the next two. This leads to natural and open questions about categorification of alternating links, and to questions on the structure of flag algebras. Joint work with Thao Vuong and Sergey Norin.

Contact person:
Vaughan Jones

October 10, 2013

Fall break

October 15, 2013

Andrei Rapinchuk (University of Virginia)

Weakly commensurable groups

The notion of weak commensurability (of Zariski-dense subgroups
of semi-simple algebraic groups) was introduced in the ongoing joint work
with Gopal Prasad on length-commensurable and isospectral locally symmetric
spaces. We were able to determine when two arithmetic subgroups are weakly
commensurable. This led to various geometric results, some of which are related to the
famous question “Can one hear the shape of a drum?”

Contact person:
Denis Osin

October 17, 2013

Faculty meeting, no colloquium

October 24, 2013

Alice Guionnet (MIT)

Maps are connected graphs which are properly embedded into a surface,
their genus is the minimal genus of such a surface. Matrix integrals have been shown to be related with the enumeration of maps since the seventies, after the work of Hooft and Brezin-Itzykson-Parisi and Zuber. This is the so-called topological expansion. Such an expansion has been used in many fields of physics and mathematics. In this talk, we shall describe this correspondence, discuss some applications and some generalizations.

Contact person:
Dietmar Bisch

November 7, 2013

Shmuel Weinberger (University of Chicago)

Towards TDA

Topological data analysis is the idea that high dimensional large data sets can sometimes be more effectively understood by looking for an underlying geometric structure which can then be exploited for the purposes of analysis. (It is one of the themes of this year’s special year at IMA.) In this talk I will try to explain some interesting problems that arise from this perspective: some good news and some bad news and then some more good news.

Contact person:
John Ratcliffe

November 11, 2013

Greta Panova (UCLA)

Combinatorics and complexity of Kronecker coefficients

Kronecker coefficients lie at the intersection of representation theory, algebraic combinatorics and, most recently, complexity theory. They count the multiplicities of irreducible representations in the tensor product of two other irreducible representations of the symmetric group. While their study was initiated almost 75 years ago, remarkably little is known about them. One of the major problems of algebraic combinatorics is to find an explicit positive combinatorial formula for these coefficients. Recently, this problem found a new meaning in the field of Geometric Complexity Theory, initiated by Mulmuley and Sohoni, where certain conjectures on the complexity of computing and deciding positivity of Kronecker coefficients are part of a program to prove the “P vs NP” problem. In this talk we will give an overview of this topic and we will describe several problems with some results on different aspects of the Kronecker coefficients. We will explore Saxl conjecture stating that the tensor square of certain irreducible representation of S_n contains every irreducible representation, and present a criterion for determining when a Kronecker coefficient is nonzero. In a more combinatorial direction, we will show how to prove certain unimodality results using Kronecker coefficients, including the classical Sylvester’s theorem on the unimodality of q-binomial coefficients (as polynomials in q). We will also present some results on complexity in light of Mulmuley’s conjectures.
The presented results are based on joint work with Igor Pak and Ernesto Vallejo.

Contact person:
Denis Osin

November 14, 2013

Ross Geoghegan (Binghamton)

Seeing geometry in certain kinds of modules

I’ll begin by explaining a very primitive notion of non-positive curved
space called a CAT(0) space. It’s so simple that it can be understood by
anyone who knows what a metric space is and who likes geometry. My CAT(0)
space M will come with a group G of isometries of M. This leads to the
notion of the limit set of this action of G. Much more interesting, and
the focus of my talk, is a set of special limit points called the
“horospherical limit set”. After a short discussion of what this means in
general I’ll explain how it shows up in several parts of real mathematics:
Fuchsian and Kleinian groups, discrete subgroups of Lie groups, and
tropical geometry, as well as, more generally, the issue of trying to see
geometry in the structure of ZG-modules. This is a colloquium talk
arranged around my joint work with Robert Bieri.

Contact person:
Mike Mihalik

November 21, 2013

Doron Lubinsky (Georgia Tech)

Pushing Polynomial Reproducing Kernels to Their Nonpolynomial Limits

Polynomial reproducing kernels are an essential tool in analyzing orthogonal polynomials and orthogonal expansions. They also play a key role in universality limits for random matrices. We discuss these connections. Moreover, we analyze how such the limiting form of these polynomial kernels becomes the sinc kernel for Paley-Wiener spaces.

Contact person:
Edward Saff

December 3, 2013

Special lecture

January 9, 2014

Special lecture

January 13, 2014

Special lecture

January 14, 2014

Special lecture

January 16, 2014

Special lecture

January 20, 2014

Special lecture

January 21, 2014

Faculty meeting, no colloquium

February 6, 2014

Ciprian Manolescu (UCLA)

The triangulation conjecture

The triangulation conjecture stated that any n-dimensional topological manifold is homeomorphic to a simplicial complex. It is true in dimensions at most 3, but false in dimension 4 by the work of Casson and Freedman. In this talk I will explain the proof that the conjecture is also false in higher dimensions. This result is based on previous work of Galewski-Stern and Matumoto, who reduced the problem to a question in low dimensions (the existence of elements of order 2 and Rokhlin invariant one in the 3-dimensional homology cobordism group). The low-dimensional question can be answered in the negative using a variant of Floer homology, Pin(2)-equivariant Seiberg-Witten Floer homology.

Contact person:
Bruce Hughes

February 13, 2014

David Ebin (Stony Brook University)

Geodesics on diffeomorphism groups and their subgroups.

In the past fifty years certain problems in both classical and modern physics have been studied from the perspective of geodesics on groups. This began with the work of V. I. Arnold on the study of motions of a rigid body and of incompressible fluids, the former using the group SO(3) and the latter using the group of diffeomorphisms of a region which preserve its volume element (volumorphisms). Subsequently, this work has been expanded to include diffeomorphisms that preserve a symplectic form, a contact form, and a contact structure (respectively, symplectoporphisms, quantomorphisms and contactomorphisms). We shall discuss the analysis required to find the geodesics in these situations. It will involve solving ODE’s on infinite dimensional spaces an some PDE techniques as well.

Contact person:
Gieri Simonett, Marcelo Disconzi

February 20, 2014

Finitely based algebras

A law is a universally quantified equation, such as the associative law
or the commutative law. I intend to talk about the problem
of determining which algebras have a finite basis for their laws.
Most of my talk will be about the laws of finite algebras.

Contact person:
Ralph McKenzie

February 27, 2014

Alex Furman (UIC)

Boundary theory and simplicity of the Lyapunov spectrum

Consider products of matrices that are chosen using some ergodic
stationary random process on $G=SL_d(R)$, e.g. a random walk on $G$.
The Multiplicative Ergodic Theorem (Oseledets) asserts that the
asymptotically such products behave as powers $\Lambda^n$ of a fixed
diagonal matrix $\Lambda$, called the Lyapunov spectrum of the system.
The spectrum $\Lambda$ depends on the system in a mysterious way, and
is almost never known explicitly.
The best understood case is that of random walks, where by the work of
Furstenberg, Guivarc’h-Raugi, and Gol’dsheid-Margulis we know that the
spectrum is simple (i.e. all values are distinct) provided the random
walk is not trapped in a proper algebraic subgroup.
Recently, Avila and Viana proved a conjecture of Kontsevich-Zorich
that asserts simplicity of the Lyapunov spectrum for another system
related to Teichmuller flow.

In the talk we shall describe an approach to proving simplicity of the
spectrum based on ideas from boundary theory that were developed to
prove rigidity of lattices.
Based on joint work with Uri Bader.

Contact person:
Jesse Peterson

March 6, 2014

Spring break, no colloquium

March 13, 2014

Mikhail Belolipetsky (IMPA)

Free subgroups of three-manifold groups

A group G is called k-free if any subgroup of G generated by k
elements is free. I will show that the fundamental groups of certain
congruence covers of hyperbolic 3-manifolds are k-free with k growing
polynomially with the degree of the cover, which is asymptotically the
fastest possible growth rate. This result resolves a special case of a
conjecture of M. Gromov. An interesting feature is that the proof of
this group theoretic result relies on geometric properties of
2-dimensional systoles. If time permits, I will also discuss some
other a priori non-geometric problems that can be solved using
systolic geometry.

Contact person:
Mark Sapir

March 20, 2014

Faculty meeting, no colloquium

March 27, 2014

Andrew Toms (Purdue)

Interpolating between classical and dynamical dimension

Motivated by a suggestion of Gromov, Lindenstrauss and Weiss
developed a dimension theory for topological dynamical systems called
the mean dimension. While it functions well for minimal systems, the
theory breaks down as periodic points become more prevalent. In this
talk I’ll introduce a notion of dimension using modules over C*-algebras
which aims to repair these defects while maintaining the character of
the mean dimension for minimal systems.

Contact person:
Jesse Peterson

April 3, 2014

Jean Bellissard (Georgia Tech)

3:10 pm, SC 1320

Transverse Geometry of Tiling Spaces

The tiling space of a given tiling will be described first on
the example of the Fibonacci sequence and of the octagonal tiling. A more
formal defintion will be provided then. It will be shown that for the
class of tilings that are repetitive, aperiodic with finite local
complexity the tiling space is a Cantor set. Hence a Geometry can only be
described in the context of Noncommutative approach, through “spectral
triples”. This concept will be defined and exemplified in the case of
Riemannian manifolds. Then the Palmer-Pearson spectral triple will be
described in detail. At last the construction of the Pearson operator, the
Cantor analog of the Laplace-Beltrami operator will be defined. Its
spectral properties will be quickly described.

Contact person:
Arnaud Brothier

April 10, 2014

Romain Tessera (ENS Lyon)

Poincaré inequalities and metric embeddings into Banach spaces

An expander is an unbounded sequence of finite graphs with very high connectivity, but uniformly bounded degree. These graphs have applications ranging from computer sciences to non-commutative geometry. However, they are not easy to construct and the first examples were actually obtained by random methods. More recently, explicit examples were produced in the context of group theory. Gromov observed that among its various strange properties, an expander cannot be “coarsely” embedded into a Hilbert space, and it was open for some time whether ”containing an expander” was the only obstruction. After recalling the relevant basic notions and results, we shall expose recent development about this question.

Contact person:
Denis Osin

April 15, 2014

Marcelo Disconzi (Vanderbilt)

The relativistic Navier-Stokes and Einstein’s equations

In this talk, we shall discuss the problem of formulating a relativistic theory of viscous fluids. After a brief introduction to the relevant concepts of General Relativity and the Einstein equations, we shall explain the origins of the problem and the known difficulties in addressing it. We finish with some of our recent results, which point toward a resolution of the problem. The talk will be accessible to non-specialists, and it will be largely self-contained. Graduate and advanced undergraduate students are encouraged to attend.

Contact person:
Dietmar Bisch

April 17, 2014

Sylvia Serfaty (NYU)

Questions of crystallization in Coulomb systems

We are interested in systems of points with Coulomb interaction. An
instance is the classical Coulomb gas, another is vortices in the
Ginzburg-Landau model of superconductivity, where one observes in certain
regimes the emergence of densely packed point vortices forming perfect
triangular lattice patterns, named Abrikosov lattices in physics. In
joint works with Etienne Sandier and with Nicolas Rougerie, we studied
both systems and derived a “Coulombian renormalized energy”. I will
present it, examine the question of its minimization and its link with
the Abrikosov lattice and weighted Fekete points. I will describe its
relation with the statistical mechanics models mentioned above and show
how it leads to expecting crystallisation in the low temperature limit.

Contact person:
Ed Saff

Colloquium Chair (2013-2014): Denis Osin

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