## Seminars and Colloquia by Series

Tuesday, April 7, 2009 - 16:00 , Location: Skiles 269 , Andrei Kapaev , Indiana University-Purdue University Indianapolis , Organizer: Stavros Garoufalidis
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.
Monday, March 30, 2009 - 15:00 , Location: Skiles 255 , Jeff Geronimo , School of Mathematics, Georgia Tech , Organizer: Jeff Geronimo
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.
Monday, February 23, 2009 - 14:00 , Location: Skiles 255 , Eric Rains , Caltech , Organizer: Plamen Iliev
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.
Wednesday, February 11, 2009 - 14:00 , Location: Skiles 255 , Giuseppe Mastroianni , Universita della Basilcata, Potenza, Italy , Organizer: Doron Lubinsky
Monday, February 2, 2009 - 14:00 , Location: Skiles 255 , Chris Heil , School of Mathematics, Georgia Tech , Organizer: Plamen Iliev
The Balian-Low Theorem is a strong form of the uncertainty principle for Gabor systems that form orthonormal or Riesz bases for L^2(R). In this talk we will discuss the Balian-Low Theorem in the setting of Schauder bases. We prove that new weak versions of the Balian-Low Theorem hold for Gabor Schauder bases, but we constructively demonstrate that several variants of the BLT can fail for Gabor Schauder bases that are not Riesz bases. We characterize a class of Gabor Schauder bases in terms of the Zak transform and product A_2 weights; the Riesz bases correspond to the special case of weights that are bounded away from zero and infinity. This is joint work with Alex Powell (Vanderbilt University).
Monday, December 1, 2008 - 14:00 , Location: Skiles 255 , Sergey Tikhonov , ICREA and CRM, Barcelona , Organizer: Michael Lacey
In this talk we will discuss a generalization of monotone sequences/functions as well as of those of bounded variation. Some applications to various problems of analysis (the Lp-convergence of trigonometric series, the Boas-type problem for the Fourier transforms, the Jackson and Bernstein inequalities in approximation, etc.) will be considered.
Wednesday, November 26, 2008 - 14:00 , Location: Skiles 255 , Yoshihiro Sawano , Gakushuin University, Japan , Organizer: Michael Lacey

Note time change.

Let I_\alpha be the fractional integral operator. The Olsen inequality, useful in certain PDEs, concerns multiplication operators and fractional integrals in the L^p-norm, or more generally, the Morrey norm. We strenghten this inequality from the one given by Olsen.
Monday, November 24, 2008 - 14:00 , Location: Skiles 255 , Ignacio Uriarte-tuero , Michigan State University , Organizer: Michael Lacey
In his celebrated paper on area distortion under planar quasiconformal mappings (Acta 1994), K. Astala proved that a compact set E of Hausdorff dimension d is mapped under a K-quasiconformal map f to a set fE of Hausdorff dimension at most d' = \frac{2Kd}{2+(K-1)d}, and he proved that this result is sharp. He conjectured (Question 4.4) that if the Hausdorff measure \mathcal{H}^d (E)=0, then \mathcal{H}^{d'} (fE)=0. This conjecture was known to be true if d'=0 (obvious), d'=2 (Ahlfors), and more recently d'=1 (Astala, Clop, Mateu, Orobitg and UT, Duke 2008.) The approach in the last mentioned paper does not generalize to other dimensions. Astala's conjecture was shown to be sharp (if it was true) in the class of all Hausdorff gauge functions in work of UT (IMRN, 2008). Finally, we (Lacey, Sawyer and UT) jointly proved completely Astala's conjecture in all dimensions. The ingredients of the proof come from Astala's original approach, geometric measure theory, and some new weighted norm inequalities for Calderon-Zygmund singular integral operators which cannot be deduced from the classical Muckenhoupt A_p theory. These results are intimately related to (not yet fully understood) removability problems for various classes of quasiregular maps. The talk will be self-contained.
Thursday, November 20, 2008 - 11:00 , Location: Skiles 255 , Dmitriy Bilyk , IAS & U South Carolina , Organizer: Michael Lacey

Note change in time.

The theory of geometric discrepancy studies different variations of the following question: how well can one approximate a uniform distribution by a discrete one, and what are the limitations that necessarily arise in such approximations. Historically, the methods of harmonic analysis (Fourier transform, Fourier series, wavelets, Riesz products etc) have played a pivotal role in the subject. I will give an overview of the problems, methods, and results in the field and discuss some latest developments.
Monday, November 10, 2008 - 14:00 , Location: Skiles 269 , Stavros Garoufalidis , School of Mathematics, Georgia Tech , Organizer: Stavros Garoufalidis
It is easy to ask for the number T(g,n) of (rooted) graphs with n edges on a surface of genus g. Bender et al gave an asymptotic expansion for fixed g and large n. The contant t_g remained missing for over 20 years, although it satisfied a complicated nonlinear recursion relation. The relation was vastly simplified last year. But a further simplification was made possible last week, thus arriving to Painleve I. I will review many trivialities and lies about this famous non-linear differential equation, from a post modern point of view.