Seminars and Colloquia by Series

Two weight inequalities for singular integrals

Series
Analysis Seminar
Time
Wednesday, August 26, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Michael Lacey – Georgia Institute of Technology
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.

Laguerre-Sobolev type orthogonal polynomials. Algebraic and analytic properties

Series
Analysis Seminar
Time
Wednesday, April 29, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Francisco Marcellan – Universidad Carlos III de Madrid

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.

Aluthge iteration of an operator

Series
Analysis Seminar
Time
Monday, April 27, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Tin Yau Tam – Department of Mathematics, Auburn University
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.

Universality via the Dbar Steepest Descent Method

Series
Analysis Seminar
Time
Wednesday, April 22, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Peter D. Miller – University of Michigan
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.

Hadamard's conjecture, Green function estimates and potential theory for higher order elliptic operators

Series
Analysis Seminar
Time
Monday, April 20, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Svitlana Mayboroda – Purdue University

Please Note: 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.

Universality Limits for Random Matrices and de Branges Spaces of Entire Functions

Series
Analysis Seminar
Time
Monday, April 13, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Doron Lubinsky – School of Mathematics, Georgia Tech
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.

Quasi-linear Stokes phenomenon for the Painleve first equation

Series
Analysis Seminar
Time
Tuesday, April 7, 2009 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Andrei Kapaev – Indiana University-Purdue University Indianapolis
Solutions of the simplest of the Painleve equations, PI, y'' = 6y^2+x, exhibit surprisingly rich asymptotic properties as x is large. Using the Riemann-Hilbert problem approach, we find an exponentially small addition to an algebraically large background admitting a power series asymptotic expansion and explain how this "beyond of all orders" term helps us to compute the coefficient asymptotics in the preceding series.

Contracted asymptotics for orthogonal polynomials with unbounded recurrence coefficients

Series
Analysis Seminar
Time
Monday, March 30, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Jeff Geronimo – School of Mathematics, Georgia Tech
The contracted asymptotics for orthogonal polynomials whose recurrence coefficients tend to infinity will be discussed. The connection between the equilibrium measure for potential problems with external fields will be exhibited. Applications will be presented which include the Wilson polynomials.

Elliptic hypergeometric integrals

Series
Analysis Seminar
Time
Monday, February 23, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Eric Rains – Caltech
Euler's beta (and gamma) integral and the associated orthogonal polynomials lie at the core of much of the theory of special functions, and many generalizations have been studied, including multivariate analogues (the Selberg integral; also work of Dixon and Varchenko), q-analogues (Askey-Wilson, Nasrallah-Rahman), and both (work of Milne-Lilly and Gustafson; Macdonald and Koornwinder for orthgonal polynomials). (Among these are the more tractable sums arising in random matrices/tilings/etc.) In 2000, van Diejen and Spiridonov conjectured a further generalization of the Selberg integral, going beyond $q$ to the elliptic level (replacing q by a point on an elliptic curve). I'll discuss two proofs of their conjecture, and the corresponding elliptic analogue of the Macdonald and Koornwinder orthogonal polynomials. In addition, I'll discuss a further generalization of the elliptic Selberg integral with a (partial) symmetry under the exceptional Weyl group E_8, and its relation to Sakai's elliptic Painlev equation.

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