## Seminars and Colloquia by Series

Thursday, October 15, 2009 - 14:00 , Location: Skiles 255 **NOTE ROOM CHANGE AND SPECIAL DAY** , Lillian Wong , University of Oklahoma , Organizer:
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.
Wednesday, October 14, 2009 - 14:00 , Location: Skiles 269 , Marcus Carlsson , Purdue University , Organizer:
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.
Wednesday, October 7, 2009 - 14:00 , Location: Skiles 269 , Ramazan Tinaztepe , Georgia Tech , Organizer: Plamen Iliev
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.
Wednesday, September 30, 2009 - 14:00 , Location: Skiles 269 , Plamen Illiev , Georgia Tech , Organizer:
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.
Wednesday, September 23, 2009 - 14:00 , Location: Skiles 269 , Maxym Yattselev , Vanderbilt University , Organizer:
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.
Wednesday, September 9, 2009 - 14:00 , Location: Skiles 269 , Shannon Bishop , Georgia Tech , Organizer:
We describe how time-frequency analysis is used to analyze boundedness and Schatten class properties of pseudodifferential operators and Fourier integral operators.
Wednesday, September 2, 2009 - 14:00 , Location: Skiles 269 , Michael Lacey , Georgia Institute of Technology , Organizer:
We will survey recent developments in the area of two weight inequalities, especially those relevant for singular integrals.  In the second lecture, we will go into some details of recent characterizations of maximal singular integrals of the speaker, Eric Sawyer, and Ignacio Uriate-Tuero.
Wednesday, August 26, 2009 - 14:00 , Location: Skiles 269 , Michael Lacey , Georgia Institute of Technology , Organizer:
We will survey recent developments in the area of two weight inequalities, especially those relevant for singular integrals.  In the second lecture, we will go into some details of recent characterizations of maximal singular integrals of the speaker, Eric Sawyer, and Ignacio Uriate-Tuero.
Wednesday, April 29, 2009 - 14:00 , Location: Skiles 255 , Francisco Marcellan , Universidad Carlos III de Madrid , Organizer: Plamen Iliev

In this contribution we study the asymptotic behaviour of polynomials orthogonal with respect to a Sobolev-Type inner product
\langle p, q\rangle_S = \int^\infty_0 p(x)q(x)x^\alpha e^{-x} dx + IP(0)^t AQ(0), \alpha > -1,
where p and q are polynomials with real coefficients,
A = \pmatrix{M_0 & \lambda\\ \lambda & M_1}, IP(0) = \pmatrix{p(0)\\ p'(0)}, Q(0) = \pmatrix{q(0)\\ q'(0)},
and A is a positive semidefinite matrix.

First, we analyze some algebraic properties of these polynomials. More precisely, the connection relations between the polynomials orthogonal with respect to the above inner product and the standard Laguerre polynomials are deduced. On the other hand, the symmetry of the multiplication operator by x^2 yields a five term recurrence relation that such polynomials satisfy.

Second, we focus the attention on their outer relative asymptotics with respect to the standard Laguerre polynomials as well as on an analog of the Mehler-Heine formula for the rescaled polynomials.

Third, we find the raising and lowering operators associated with these orthogonal polynomials. As a consequence, we deduce the holonomic equation that they satisfy. Finally, some open problems will be considered.

Monday, April 27, 2009 - 15:00 , Location: Skiles 255 , Tin Yau Tam , Department of Mathematics, Auburn University , Organizer:
Let A be a Hilbert space operator. If A = UP is the polar decomposition of A, and 0 < \lambda < 1, the \lambda-Aluthge transform of A is defined to be the operator \Delta_\lambda = P^\lambda UP^{1-\lambda}. We will discuss the recent progress on the convergence of the iteration. Infinite and finite dimensional cases will be discussed.