## Seminars and Colloquia by Series

Wednesday, September 30, 2009 - 14:00 , Location: Skiles 269 , Plamen Illiev , Georgia Tech , Organizer: Brett Wick
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: Brett Wick
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: Brett Wick
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: Brett Wick
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: Brett Wick
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.
Wednesday, April 22, 2009 - 14:00 , Location: Skiles 255 , Peter D. Miller , University of Michigan , Organizer: Jeff Geronimo
We will discuss a new method of asymptotic analysis of matrix-valued Riemann-Hilbert problems that involves dispensing with analyticity in favor of measured deviation therefrom. This method allows the large-degree analysis of orthogonal polynomials on the real line with respect to varying nonanalytic weights with external fields having two Lipschitz-continuous derivatives, as long as the corresponding equilibrium measure has typical support properties. Universality of local eigenvalue statistics of unitary-invariant ensembles in random matrix theory follows under the same conditions. This is joint work with Ken McLaughlin.
Monday, April 20, 2009 - 15:00 , Location: Skiles 255 , Svitlana Mayboroda , Purdue University , Organizer: Michael Lacey

Note special time

In 1908 Hadamard conjectured that the biharmonic Green function must be positive. Later on, several counterexamples to Hadamard's conjecture have been found and a variety of upper estimates were obtained in sufficiently smooth domains. However, the behavior of the Green function in general domains was not well-understood until recently. In a joint work with V. Maz'ya we derive sharp pointwise estimates for the biharmonic and, more generally, polyharmonic Green function in arbitrary domains. Furthermore, we introduce the higher order capacity and establish an analogue of the Wiener criterion describing the precise correlation between the geometry of the domain and the regularity of the solutions to the polyharmonic equation.
Monday, April 13, 2009 - 14:00 , Location: Skiles 255 , Doron Lubinsky , School of Mathematics, Georgia Tech , Organizer: Plamen Iliev
It turns out that the sinc kernel is not the only kernel that arises as a universality limit coming from random matrices associated with measures with compact support. Any reproducing kernel for a de Branges space that is equivalent to a Paley-Winer space may arise. We discuss this and some other results involving de Branges spaces, universality, and orthogonal polynomials.