College of Arts and Science Vanderbilt University
Vanderbilt Univeristy College of Arts and Science

Colloquium, Academic Year 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)

About topological expansions

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  
Keith Kearnes (University of Colorado)

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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