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Mathematics & Statistics

# Colloquia — Spring 2009

## Friday, April 10, 2009

Title
Speaker

Time
Place

A remark on a theorem of Jech and Shelah
Peter Komjath
Emory University
3:00pm-4:00pm
PHY 109
Vilmos Totik

Abstract

Jech and Shelah showed that if $$\aleph_{\omega}$$ is a strong limit and $$2^{\aleph_{\omega}}\aleph_{\omega_{1}}$$ holds (the consistency of this statement is unknown) then a system of subsets of $$\omega_1$$ with certain properties exist. They proved the rather surprising fact that such a system exists anyway.

We give a simpler proof of the latter result.

## Friday, March 6, 2009

Title
Speaker

Time
Place

Exact Solutions to Integrable Evolution Equations
Tuncay Aktosun
University of Texas at Arlington
3:00pm-4:00pm
PHY 109
Wen-Xiu Ma

Abstract

A method is presented to construct certain exact solutions to those nonlinear partial differential equations integrable by the inverse scattering transform with the help of a Marchenko integral equation. Explicit formulas are obtained to express such exact solutions in a compact form in terms of matrix exponentials. Such exact solutions can alternatively be written explicitly as algebraic combinations of exponential, trigonometric, and polynomial functions of the spatial and temporal coordinates. The method is applied to some integrable equations such as the Korteweg-de Vries equation on the half line and the nonlinear Schröedinger equation.

## Friday, February 27, 2009

Title
Speaker

Time
Place

An equivalence of Ward's bound and its applications
Xiaoyu Liu
Wright State University
3:00pm-4:00pm
PHY 109
Xiang-dong Hou

Abstract

In 1981, Harold N. Ward introduced the concept of divisibility in linear codes and investigated linear codes from this point of view. Briefly speaking, divisible codes are simply codes whose codewords all have weights divisible by a nontrivial integer, say $$\Delta$$ (called a divisor of the code). The reason that these codes are interesting is that people observe nontrivial divisibility in most of the optimal codes, as well as in many other good codes. Ward proved a bound on the dimension of a linear divisible code over a finite field. In this talk, we will show an equivalence of this Ward's bound. Using the equivalence, we generalize Ward's bound to some nonlinear codes. Another application of the result is to generalize the Gleason-Pierce-Ward theorem to additive codes.

## Friday, February 20, 2009

Title
Speaker

Time
Place

A Short Survey of Extension Theorems
Robb Fry
Thomson Rivers University
3:00pm-4:00pm
PHY 109
Lesław Skrzypek

Abstract

We shall briefly examine extension theorems in various contexts, as well as a few of their applications. As classical examples we include Tietze's Theorem, Lavrentieff's Theorem, andthe Hahn-Banach Theorem. Recent and less well known results concerning extensions of locally uniformly convex norms and extensions of real-valued, smooth functions on Banach spaces will also be examined.

## Friday, February 13, 2009

Title
Speaker

Time
Place

Bilinear Forms on the Dirichlet Space
Brett Wick
University of South Carolina
3:00pm-4:00pm
PHY 109
Dmitry Khavinson

Abstract

The Dirichlet space is the set of analytic functions on the disc that have a square integrable derivative. In this talk we will discuss necessary and sufficient conditions in order to have a bilinear form on the Dirichlet space be bounded. This condition will be expressed in terms of a Carleson measure condition for the Dirichlet space. One can view this result as the Dirichlet space analogue of Nehari's Theorem for the classical Hardy space on the disc. This talk is based on joint work with N. Arcozzi, R. Rochberg, and E. Sawyer.

## Friday, February 6, 2009

Title
Speaker

Time
Place

Non-intersecting random paths and multiple orthogonal polynomials
Andrei Martínez-Finkelshtein
Almería, SPAIN
3:00pm-4:00pm
PHY 109
E. Rakhmanov

Abstract

Determinantal point processes are of considerable current interest in probability theory and mathematical physics, since they arise naturally in random matrix theory, non-intersecting paths, certain combinatorial and stochastic growth models and representation theory of large groups. We are interested in a situation when the determinantal point process is related to polynomials of multiple orthogonality (that is, when the orthogonality conditions are distributed among several weights). The simplest case is of the non-intersecting paths of particles performing brownian motion, ending at two or more fixed positions (this model is connected also with the eigenvalue distribution of GUE with external source).

We study $$n$$ non-intersecting squared Bessel processes in the confluent case: all particles start at time $$t=0$$ at the same positive value $$x=a$$, remain positive (due to the nature of the squared Bessel process) and are conditioned to end at time $$t=T$$ at $$x=0$$. In the limit $$n\to\infty$$, after appropriate rescaling, the paths fill out a region in the $$tx$$-plane. In particular, the paths initially stay away from the hard edge at $$x=0$$, but at a certain critical time the smallest paths hit the hard edge and from then on are stuck to it. The phase transition at the critical time is a new feature of the present model.

A key fact in the analysis is that in the confluent case the biorthogonal ensemble reduces to a multiple orthogonal polynomial ensemble, corresponding to a Nikishin system of two modified Bessel-type weights. This means that there is a $$3\times 3$$ RH problem characterizing this model, that can be analyzed in the large $$n$$ limit using the Deift-Zhou steepest descent method.

This is a joint work with A. B. J. Kuijlaars and F. Wielonsky.

## Friday, January 30, 2009

Title
Speaker

Time
Place

TBA
Alfonso Montes Rodríguez
Sevilla, SPAIN
4:00pm-5:00pm
PHY 109
Sherwin Kouchekian

Abstract

TBA

## Wednesday, January 21, 2009

Title
Speaker

Time
Place

Fine asymptotics for Bergman orthogonal polynomials over domains with corners
Nikos Stylianopoulos
University of Cyprus
3:00pm-4:00pm
NES 104
Dmitry Khavinson

Abstract

Let $$G$$ be a bounded simply-connected domain in the complex plane $$\mathbb{C}$$, whose boundary $$\Gamma:=\partial G$$ is a Jordan curve, and let $$\{p_n\}^{\infty}_{n=0}$$ denote the sequence of Bergman polynomials of $$G$$. This is defined as the sequence $$p_n(z)=\lambda_nz^n+\dotsb,\quad \lambda_n > 0,\quad n=0,1,2,\dotsc,$$ of polynomials that are orthonormal with respect to the inner product $$\langle f,g\rangle:=\int_G f(z)g(z)\,dm(z),$$ where $$dm$$ stands for the area measure.

The purpose of the talk is to report on recent results regarding the fine asymptotic behaviour of the polynomials $$p_n(z)$$, in $$\Omega:=\mathbb{C}\setminus(G)$$, and the leading coefficients $$\lambda_n$$, $$n\in\mathbb{N}$$, in cases when the boundary $$\Gamma$$ is piecewise analytic without cusps. These results complement an investigation started in the 1920's by T. Carleman, who obtained the fine asymptotics for domains with analytic boundaries and carried over by P. K. Suetin in the 1960's, who established them for domains with smooth boundaries.

## Friday, January 16, 2009

Title
Speaker

Time
Place
For a $$1$$-dimensional link, the crossing number is defined by the minimal number of crossings of any diagram. For $$2$$-dimensional links (closed surfaces embedded in $$4$$-space), minimal triple point numbers correspond to crossing numbers of $$1$$-dimensional links. In this talk, we introduce a method to estimate minimal triple point numbers with symmetric quandles, and we show some results which can be obtained by our method.