# THEMATIC PROGRAMS

March 29, 2017

## THEMATIC PROGRAM ON NEW TRENDS IN HARMONIC ANALYSIS

### Working and Research Seminars January - April 2008

 Graduate Courses Back to Thematic Year home

UPCOMING SEMINARS
Tuesdays & Thursdays,

 May 8 2-3 p.m. library David Damanik (Rice University) The repetition property and applications to spectral theory We present joint work with Michael Boshernitzan on a fairly weak repetition property of a topological dynamical system, which implies for generic sampling functions a property for their orbits, which was introduced by Alexander Gordon in 1976 to exclude eigenvalues of the associated Schr"odinger operators. A particular consequence is that a typical skew-shift model with a generic sampling function has no eigenvalues for almost every element in the hull. Apr. 4 2-3 p.m. library Vitaly Bergelson (Ohio State University) Van der Corput trick and its ramifications. Let (x_n) be a sequence of real numbers. Van der Corput's difference theorem states that if for every k in N the sequence x_(n+k)-x_n, n=1,2,... is uniformly distributed mod1, then the sequence (x_n) also is uniformly distributed mod1. For example, the uniform distribution of the sequence (n^2a mod1), where a is an irrational number, is an immediate consequence of this theorem. We will discuss various refinements of this result which lead to interesting applications in ergodic theory and combinatorics. Apr. 3 2-3 p.m. (Rm 230 or library) Vitaly Bergelson (Ohio State University) Applications of Poincare recurrence theorem to combinatorics and number theory. Poincare recurrence theorem, one of the earliest results in ergodic theory, was introduced in 1890 by Henri Poincare in his famous Acta Mathematica paper on the stability of solar system. The goal of this lecture is to discuss some modern applications of this classical theorem to problems in combinatorics, algebra and number theory. Mar. 27 1:00 p.m. Library Ilya Shkredov 3-D corners Mar. 25 1:00 p.m. Library Ilya Shkredov Theorem on 2-D corners Mar. 20 1:00 p.m. Library Michael Lacey Progressions of Length 3 in finite fields. Roth-Meschulum argument Feb. 7 Serban Costea Sobolev capacity and Hausdorff measures in metric measure spaces Feb. 5 1 pm Tuesday: Michael Lacey Lipschitz Kakeya Maximal Function 2 pm Tuesday: Daryl Geller Heat Trace Asymptotics, Spherical Wavelets and Dark Energy This is joint work with Azita Mayeli. The heat trace asymptotics on the sphere $S^n$ are known explicitly. (That is, if $\Delta$ is the Laplace-Beltrami operator on the sphere, one knows the numbers $a_m$ for which $\mbox{tr}(e^{-t\Delta}) \sim \sum_{m=0}^{\infty} t^{(m-n)/2}a_m$. I. Polterovich recently gave a very explicit expression for them.) On the sphere, the heat kernel $K_t(x,y) := g_t(x \cdot y)$ is a function of $x \cdot y$. Using the heat trace asymptotics we show how one can compute the Maclaurin series for $4\pi g_t(\cos \theta)$. For small $t$ this gives rise to what appears to be an excellent approximation to the heat kernel, $4\pi g_t(\cos \theta) \sim \frac{e^{-\theta^2/4t}}{s}[(1+\frac{t}{3}+\frac{t^2}{15}+\frac{4t^3}{315}+\frac{t^4}{315}) + \frac{\theta^2}{4}(\frac{1}{3}+\frac{2t}{15}+\frac{4t^2}{105}+\frac{4t^3}{315})],$ on $S^2$. Using this approximation we are able to approximately evaluate new wavelets for the sphere, which we have devised, and which have significant advantages over those currently in use. Such wavelets are being used by teams of statisticians and astrophysicists to analyze cosmic microwave background radiation, in particular to estimate the percentage of dark energy in the universe and its properties. (We have been invited to visit one of these teams in March, in Rome.) Jan. 24 Brett Wick (University of South Carolina) Bounded analytic projections Jan. 29 1 pm Tuesday: Alex Iosevich (University of Missouri) Sums and products in finite fields via higher dimensional geometry and Fourier analysis 2 pm Tuesday Hrant Hakobyan (University of Toronto) Conformal dimension: Cantor sets and curve families. The infimal Hausdorff dimension of all quasysimmetric (qs) images of a metric space X is called Conformal dimension of X. A subset X of a line is called qs thick if every qs self map of the line maps X to a set of positive measure. Bishop and Tyson asked if there are sets which are not qs thick but still have conformal dimension 1. We will answer this affirmatively by proving that many Cantor sets of Hausdorff dimension 1 have also Conformal dimension 1. Very often the obstruction for minimizing the dimension of a space X is the existence of "sufficiently many" curves in X. If the time permits we will define a dimension type quasiconformal invariant for a curve family in the plane. We will discuss some examples and will illustrate some connections of this invariant with the Conformal dimension of X. Jan. 24 Thomas Hytonen Thoughts about tent spaces Jan. 22 Ignacio Uriarte-Tuero Removability problems for bounded, BMO and H\"{o}lder quasiregular mappings A classical problem in complex analysis is to characterize the removable sets for various classes of analytic functions: H\"{o}lder, Lipschitz, BMO, bounded (this last case gives rise to the analytic capacity and the Painlev\'{e} problem which has been recently solved by Tolsa.) One can ask the same questions in the setting of K-quasiregular maps (since they are a K-quasiconformal map followed by an analytic map.) Most of the bounded case was dealt with in a joint paper with K. Astala, A. Clop, J.Mateu and J.Orobitg, [ACMOUT]. The BMO case was dealt with in [ACMOUT] except for a gap at the critical dimension (Question 4.2 in [ACMOUT].) I answered the question filling the gap in [UT]. The Lipschitz case was dealt with by A. Clop, as well as most of the H\"{o}lder case, where again a gap at the critical dimension was left. In a joint paper with A. Clop [CUT] we closed the gap. I will summarize the results and give some ideas of the proofs in the above papers. The talk will be self-contained. References: [ACMOUT] Kari Astala, Albert Clop, Joan Mateu, Joan Orobitg and Ignacio Uriarte-Tuero. Distortion of Hausdorff measures and improved Painlev\'{e} removability for bounded quasiregular mappings. Duke Math J., to appear. [CUT] Albert Clop and Ignacio Uriarte-Tuero. Sharp Nonremovability Examples for H\"{o}lder continuous quasiregular mappings in the plane. Preprint. [UT] Ignacio Uriarte-Tuero. Sharp Examples for Planar Quasiconformal Distortion of Hausdorff Measures and Removability. Submitted.