Seminars and Colloquia by Series

On a Bargmann transform and coherent states for the n-sphere

Series
Analysis Seminar
Time
Wednesday, November 4, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Dr Carlos Villegas BlasInstituto de Matematicas UNAM, Unidad. Cuernavaca
We will introduce a Bargmann transform from the space of square integrable functions on the n-sphere onto a suitable Hilbert space of holomorphic functions on a null quadric. On base of our Bargmann transform, we will introduce a set of coherent states and study their semiclassical properties. For the particular cases n=2,3,5, we will show the relation with two known regularizations of the Kepler problem: the Kustaanheimo-Stiefel and Moser regularizations.

The Extremal Nevanlinna-Pick problem for Riemann Surfaces

Series
Analysis Seminar
Time
Wednesday, October 28, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Mrinal RagupathiVanderbilt University
Given points $z_1,\ldots,z_n$ on a finite open Riemann surface $R$ and complex scalars $w_1,\ldots,w_n$, the Nevanlinna-Pick problem is to determine conditions for the existence of a holomorphic map $f:R\to \mathbb{D}$ such that $f(z_i) = w_i$. In this talk I will provide some background on the problem, and then discuss the extremal case. We will try to discuss how a method of McCullough can be used to provide more qualitative information about the solution. In particular, we will show that extremal cases are precisely the ones for which the solution is unique.

From transfinite diameter to order-density to best-packing: the asymptotics of ground state configurations

Series
Analysis Seminar
Time
Friday, October 23, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Doug HardinVanderbilt University
I will review recent and classical results concerning the asymptotic properties (as N --> \infty) of 'ground state' configurations of N particles restricted to a d-dimensional compact set A\subset {\bf R}^p that minimize the Riesz s-energy functional \sum_{i\neq j}\frac{1}{|x_{i}-x_{j}|^{s}} for s>0. Specifically, we will discuss the following (1) For s < d, the ground state configurations have limit distribution as N --> \infty given by the equilibrium measure \mu_s, while the first order asymptotic growth of the energy of these configurations is given by the 'transfinite diameter' of A. (2) We study the behavior of \mu_s as s approaches the critical value d (for s\ge d, there is no equilibrium measure). In the case that A is a fractal, the notion of 'order two density' introduced by Bedford and Fisher naturally arises. This is joint work with M. Calef. (3) As s --> \infty, ground state configurations approach best-packing configurations on A. In work with S. Borodachov and E. Saff we show that such configurations are asymptotically uniformly distributed on A.

Inequalities for Derivatives and their Applications

Series
Analysis Seminar
Time
Wednesday, October 21, 2009 - 14:00 for 8 hours (full day)
Location
Skiles 269
Speaker
Yuliya BabenkoSam Houston State University
In this talk we will discuss Kolmogorov and Landau type inequalities for the derivatives. These are the inequalities which estimate the norm of the intermediate derivative of a function (defined on an interval, R_+, R, or their multivariate analogs) from some class in terms of the norm of the function itself and norm of its highest derivative. We shall present several new results on sharp inequalities of this type for special classes of functions (multiply monotone and absolutely monotone) and sequences. We will also highlight some of the techniques involved in the proofs (comparison theorems) and discuss several applications.

Point Perturbation and Asymptotics Orthogonal Polynomials

Series
Analysis Seminar
Time
Thursday, October 15, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255 **NOTE ROOM CHANGE AND SPECIAL DAY**
Speaker
Lillian WongUniversity of Oklahoma
In this talk, I will discuss some results obtained in my Ph.D. thesis. First, the point mass formula will be introduced. Using the formula, we shall see how the asymptotics of orthogonal polynomials relate to the perturbed Verblunsky coefficients. Then I will discuss two classes of measures on the unit circle -- one with Verblunsky coefficients \alpha_n --> 0 and the other one with \alpha_n --> L (non-zero) -- and explain the methods I used to tackle the point mass problem involving these measures. Finally, I will discuss the point mass problem on the real line. For a long time it was believed that point mass perturbation will generate exponentially small perturbation on the recursion coefficients. I will demonstrate that indeed there is a large class of measures such that that proposition is false.

Positive Matrices and Applications to Hankel Operators

Series
Analysis Seminar
Time
Wednesday, October 14, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Marcus CarlssonPurdue University
Given an "infinite symmetric matrix" W we give a simple condition, related to the shift operator being expansive on a certain sequence space, under which W is positive. We apply this result to AAK-type theorems for generalized Hankel operators, providing new insights related to previous work by S. Treil and A. Volberg. We also discuss applications and open problems.

Embeddings of Modulation Spaces into BMO

Series
Analysis Seminar
Time
Wednesday, October 7, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Ramazan TinaztepeGeorgia Tech
Modulation spaces are a class of Banach spaces which provide a quantitative time-frequency analysis of functions via the Short-Time Fourier Transform. The modulation spaces are the "right" spaces for time-frequency analysis andthey occur in many problems in the same way that Besov Spaces are attached to wavelet theory and issues of smoothness. In this seminar, I will talk about embeddings of modulation Spaces into BMO or VMO (the space of functions of bounded or vanishing mean oscillation, respectively ). Membership in VMO is central to the Balian-Low Theorem, which is a cornerstone of time-frequency analysis.

Trigonometric Grassmannian and a difference W-algebra

Series
Analysis Seminar
Time
Wednesday, September 30, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Plamen IllievGeorgia Tech
The trigonometric Grassmannian parametrizes specific solutions of the KP hierarchy which correspond to rank one solutions of a differential-difference bispectral problem. It can be considered as a completion of the phase spaces of the trigonometric Calogero-Moser particle system or the rational Ruijsenaars-Schneider system. I will describe the characterization of this Grassmannian in terms of representation theory of a suitable difference W-algebra. Based on joint work with L. Haine and E. Horozov.

Convergent Interpolation to Cauchy Integrals of Jacobi-type Weights and RH∂-Problems

Series
Analysis Seminar
Time
Wednesday, September 23, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Maxym YattselevVanderbilt University
We consider multipoint Padé approximation to Cauchy transforms of complex measures. First, we recap that if the support of a measure is an analytic Jordan arc and if the measure itself is absolutely continuous with respect to the equilibrium distribution of that arc with Dini-continuous non-vanishing density, then the diagonal multipoint Padé approximants associated with appropriate interpolation schemes converge locally uniformly to the approximated Cauchy transform in the complement of the arc. Second, we show that this convergence holds also for measures whose Radon–Nikodym derivative is a Jacobi weight modified by a Hölder continuous function. The asymptotics behavior of Padé approximants is deduced from the analysis of underlying non–Hermitian orthogonal polynomials, for which the Riemann–Hilbert–∂ method is used.

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