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  Colloquium Archive

Colloquia — Spring 2015

Colloquia — Fall 2014

Monday, November 24, 2014

Title

Speaker


Time
Place
Sponsor

Random Matrix Models, Non-intersecting random paths, and the Riemann-Hilbert Analysis
Andrei Martínez-Finkelshtein
Universidad de Almería
Almería, SPAIN
3:00pm-4:00pm
NES 103
E. A. Rakhmanov












Abstract

Random matrix theory (RMT) is a very active area of research and a great source of exciting and challenging problems for specialists in many branches of analysis, spectral theory, probability and mathematical physics. The analysis of the eigenvalue distribution of many random matrix ensembles leads naturally to the concepts of determinantal point processes and to their particular case, biorthogonal ensembles, when the main object to study, the correlation kernel, can be written explicitly in terms of two sequences of mutually orthogonal functions.

Another source of determinantal point processes is a class of stochastic models of particles following non-intersecting paths. In fact, the connection of these models with the RMT is very tight: the eigenvalues of the so-called Gaussian Unitary Ensemble (GUE) and the distribution of random particles performing a Brownian motion, departing and ending at the origin under condition that their paths never collide are, roughly speaking, statistically identical.

A great challenge is the description of the detailed asymptotics of these processes when the size of the matrices (or the number of particles) grows infinitely large. This is needed, for instance, for verification of different forms of “universality” in the behavior of these models. One of the rapidly developing tools, based on the matrix Riemann-Hilbert characterization of the correlation kernel, is the associated non-commutative steepest descent analysis of Deift and Zhou.

Without going into technical details, some ideas behind this technique will be illustrated in the case of a model of squared Bessel nonintersecting paths.

Friday, November 21, 2014

Title
Speaker

Time
Place
Sponsor

TBA
Marco Bertola
Concordia University
3:00pm-4:00pm
CMC 130
Seung-Yeop Lee

Abstract

TBA

Friday, November 14, 2014

Title
Speaker

Time
Place
Sponsor

TBA
Said Sidki
Universidade de Brasilia
3:00pm-4:00pm
CMC 130
Dmytro Savchuk

Abstract

TBA

Friday, November 7, 2014

Title
Speaker

Time
Place
Sponsor

TBA
Jen-Hsu Chang
UC Riverside and ?
3:00pm-4:00pm
CMC 130
Wen-Xiu Ma

Abstract

TBA

Friday, October 17, 2014

Title
Speaker



Time
Place
Sponsor

Geometric curve flows and integrable systems
Stephen Anco
Department of Mathematics and Statistics
Brock University
Ontario, CANADA
3:00pm-4:00pm
CMC 130
Wen-Xiu Ma

Abstract

The modern theory of integrable soliton equations displays many deep links to differential geometry, particularly in the study of geometric curve flows by moving-frame methods.

I will first review an elegant geometrical derivation of the integrability structure for two important examples of soliton equations: the nonlinear Schrödinger (NLS) equation; and the modified Korteweg-de Vries (mKdV) equation. This derivation is based on a moving-frame formulation of geometric curve flows which are mathematical models of vortex filaments and vortex-patch boundaries arising in ideal fluid flow in two and three dimensions. Key mathematical tools are the Cartan structure equations of Frenet frames and the Hasimoto transformation relating invariants of a curve to soliton variables, as well as the theory of Poisson brackets for Hamiltonian PDEs.

I will then describe a broad generalization of these results to geometric curve flows in semi-simple Klein geometries \(M=G/H\), giving a geometrical derivation of group-invariant (multi-component) versions of mKdV and NLS soliton equations along with their full integrability structure.

Friday, October 3, 2014

Title
Speaker

Time
Place
Sponsor

Ordering free groups and free products
Zoran Šunić
Texas A&M University
3:00pm-4:00pm
CMC 130
Milé Krajčevski

Abstract

We utilize a criterion for the existence of a free subgroup acting freely on at least one of its orbits to construct such actions of the free group on the circle and on the line, leading to orders on free groups that are particularly easy to state and work with.

We then switch to a restatement of the orders in terms of certain quasi-characters of free groups, from which properties of the defined orders may be deduced (some have positive cones that are context-free, some have word reversible cones, some of the orders extend the usual lexicographic order, and so on).

Finally, we construct total orders on the vertex set of an oriented tree. The orders are based only on up-down counts at the interior vertices and the edges along the unique geodesic from a given vertex to another. As an application, we provide a short proof of Vinogradov´s result that the free product of left-orderable groups is left-orderable.

Friday, September 26, 2014

Title
Speaker

Time
Place
Sponsor

Recent developments in Quantum invariants of knots
Mustafa Hajij
Louisiana State University
3:00pm-4:00pm
CMC 130
Mohamed Elhamdadi

Abstract

Quantum knot invariants deeply connect many domains such as lie algebras, quantum groups, number theory and knot theory. I will talk about a particular quantum invariant called the colored Jones polynomial and some of the recent work that has been done to understand it. This invariant takes the form a sequence of Laurent polynomials. I will explain how the coefficients of this sequence stabilize for certain class of knots called alternating knots. Furthermore, I will show that this leads naturally to interesting connections with number theory.

Friday, September 12, 2014

Title
Speaker

Time
Place
Sponsor

The valence of polynomial harmonic mappings
Erik Lundberg
Florida Atlantic University
3:00pm-4:00pm
CMC 130
Dmitry Khavinson

Abstract

While working to extend the Fundamental Theorem of Algebra, A. S. Wilmshurst used Bezout’s theorem to give an upper bound for the number of zeros of a (complex valued) harmonic polynomial. Although the bound is sharp in general, Wilmshurst conjectured that Bezout’s bound can be refined dramatically. Using holomorphic dynamics, the conjecture was confirmed by D. Khavinson and G. Swiatek in the special case when the anti-analytic part is linear. We will discuss recent counterexamples to other cases as well as an alternative probabilistic approach to the problem.