Seminars and Colloquia by Series

Gabor Schauder bases and the Balian-Low Theorem

Series
Analysis Seminar
Time
Monday, February 2, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Chris HeilSchool of Mathematics, Georgia Tech
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).

A general monotonicity concept and its applications in harmonic analysis and approximation theory

Series
Analysis Seminar
Time
Monday, December 1, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Sergey TikhonovICREA and CRM, Barcelona
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.

A note on Olsen inequality

Series
Analysis Seminar
Time
Wednesday, November 26, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Yoshihiro SawanoGakushuin University, Japan

Please Note: 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.

Astala's conjecture on Hausdorff measure distortion under planar quasiconformal mappings

Series
Analysis Seminar
Time
Monday, November 24, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Ignacio Uriarte-tueroMichigan State University
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.

Geometric Discrepancy and Harmonic Analysis

Series
Analysis Seminar
Time
Thursday, November 20, 2008 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Dmitriy BilykIAS & U South Carolina

Please Note: 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.

Trivialities, truths and lies about cubic graphs and Painleve I

Series
Analysis Seminar
Time
Monday, November 10, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Stavros GaroufalidisSchool of Mathematics, Georgia Tech
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.

Schatten p-class Pseudodifferential and Affine Pseudodifferential Operators

Series
Analysis Seminar
Time
Monday, November 3, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Shannon BishopSchool of Mathematics, Georgia Tech
Pseudodifferential operators and affine pseudodifferential operators arise naturally in the study of wireless communications. We discuss the origins of these operators and give new conditions on the kernels and symbols of pseudodifferential and affine pseudodifferential operators which ensure the operators are trace class (and more generally, Schatten p-class).

Bilinear Forms on the Dirichlet Space

Series
Analysis Seminar
Time
Monday, October 27, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Brett WickUniversity of South Carolina
The Dirichlet space is the set of analytic functions on the disc that have a square integrable derivative. In this talk we will discuss necessary and sufficient conditions in order to have a bilinear form on the Dirichlet space be bounded. This condition will be expressed in terms of a Carleson measure condition for the Dirichlet space. One can view this result as the Dirichlet space analogue of Nehari's Theorem for the classical Hardy space on the disc. This talk is based on joint work with N. Arcozzi, R. Rochberg, and E. Sawyer

Conformal dimension of self-affine sets and modulus of a system of measures

Series
Analysis Seminar
Time
Monday, October 6, 2008 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Hrant HakobyanUniversity of Toronto
A mapping F between metric spaces is called quasisymmetric (QS) if for every triple of points it distorts their relative distances in a controlled fashion. This is a natural generalization of conformality from the plane to metric spaces. In recent times much work has been devoted to the classification of metric spaces up to quasisymmetries. One of the main QS invariants of a space X is the conformal dimension, i.e the infimum of the Hausdorff dimensions of all spaces QS isomorphic to X. This invariant is hard to find and there are many classical fractals such as the standard Sierpinski carpet for which conformal dimension is not known. Tyson proved that if a metric space has sufficiently many curves then there is a lower bound for the conformal dimension. We will show that if there are sufficiently many thick Cantor sets in the space then there is a lower bound as well. "Sufficiently many" here is in terms of a modulus of a system of measures due to Fuglede, which is a generalization of the classical conformal modulus of Ahlfors and Beurling. As an application we obtain a new lower bound for the conformal dimension of self affine McMullen carpets.

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