Seminars and Colloquia by Series

Fuglede's spectral-set conjecture.

Series
Analysis Seminar
Time
Wednesday, December 5, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rachel Greenfeld – Bar Ilan University
A set $\Omega\subset \mathbb{R}^d$ is called spectral if the space $L^2(\Omega)$ admits an orthogonal basis of exponential functions. Back in 1974 B. Fuglede conjectured that spectral sets could be characterized geometrically by their ability to tile the space by translations. Although since then the subject has been extensively studied, the precise connection between spectrality and tiling is still a mystery.>In the talk I will survey the subject and discuss some recent results, joint with Nir Lev, where we focus on the conjecture for convex polytopes.

The fractal uncertainty principle

Series
Analysis Seminar
Time
Wednesday, November 28, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rui Han – Georgia Tech
Recently Bourgain and Dyatlov proved a fractal uncertainty principle (FUP), which roughly speaking says a function in $L^2(\mathbb{R})$ and its Fourier transform can not be simultaneously localized in $\delta$-dimensional fractal sets, $0<\delta<1$. In this talk, I will discuss a joint work with Schlag, where we obtained a higher dimensional version of the FUP. Our method combines the original approach by Bourgain and Dyatlov, in the more quantitative rendition by Jin and Zhang, with Cantan set techniques.

Cotlar’s identity for Hilbert transforms---old and new stories.

Series
Analysis Seminar
Time
Wednesday, November 14, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tao Mei – Baylor University –
Cotlar’s identity provides an easy (maybe the easiest) argument for the Lp boundedness of Hilbert transforms. E. Ricard and I discovered a more flexible version of this identity, in the recent study of the boundedness of Hilbert transforms on the free groups. In this talk, I will try to introduce this version of Cotlar’s identity and the Lp Fourier multipliers on free groups.

Portraits of RIFs: their singularities and unimodular level sets on T^2

Series
Analysis Seminar
Time
Wednesday, November 7, 2018 - 10:14 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kelly Bickel – Bucknell University
This talk concerns two-variable rational inner functions phi with singularities on the two-torus T^2, the notion of contact order (and related quantities), and its various uses. Intuitively, contact order is the rate at which phi’s zero set approaches T^2 along a coordinate direction, but it can also be defined via phi's well-behaved unimodular level sets. Quantities like contact order are important because they encode information about the numerical stability of phi, for example when it belongs to Dirichlet-type spaces and when its partial derivatives belong to Hardy spaces. The unimodular set definition is also useful because it allows one to “see” contact order and in some sense, deduce numerical stability from pictures. This is joint work with James Pascoe and Alan Sola.

Integral geometric regularity

Series
Analysis Seminar
Time
Wednesday, October 31, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joe Fu – UGA –
The centerpiece of the subject of integral geometry, as conceived originally by Blaschke in the 1930s, is the principal kinematic formula (PKF). In rough terms, this expresses the average Euler characteristic of two objects A, B in general position in Euclidean space in terms of their individual curvature integrals. One of the interesting features of the PKF is that it makes sense even if A and B are not smooth enough to admit curvatures in the classical sense. I will describe the state of our understanding of the regularity needed to make it all work, and state some conjectures that would extend it.

On the fifth Busemann-Petty problem

Series
Analysis Seminar
Time
Wednesday, October 24, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dmirty Ryabogin – Kent State University –
In 1956, Busemann and Petty posed a series of questions about symmetric convex bodies, of which only the first one has been solved.Their fifth problem asks the following.Let K be an origin symmetric convex body in the n-dimensional Euclidean space and let H_x be a hyperplane passing through the origin orthogonal to a unit direction x. Consider a hyperplane G parallel to H_x and supporting to K and let C(K,x)=vol(K\cap H_x)dist (0, G). (proportional to the volume of the cone spanned by the secion and the support point). If there exists a constant C such that for all directions x we have C(K,x)=C, does it follow that K is an ellipsoid?We give an affirmative answer to this problem for bodies sufficiently close to the Euclidean ball in the Banach Mazur distance.This is a joint work with Maria Alfonseca, Fedor Nazarov and Vlad Yaskin.

Dynamical sampling

Series
Analysis Seminar
Time
Wednesday, October 17, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Longxiu Huang – Vanderbilt University
Dynamical sampling is a new area in sampling theory that deals with signals that evolve over time under the action of a linear operator. There are lots of studies on various aspects of the dynamical sampling problem. However, they all focus on uniform discrete time-sets $\mathcal T\subset\{0,1,2,\ldots, \}$. In our study, we concentrate on the case $\mathcal T=[0,L]$. The goal of the present work is to study the frame property of the systems $\{A^tg:g\in\mathcal G, t\in[0,L] \}$. To this end, we also characterize the completeness and Besselness properties of these systems.

The Mikhlin-H\"ormander multiplier theorem: some recent developments

Series
Analysis Seminar
Time
Wednesday, October 10, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Lenka Slavikova – University of Missouri –
In this talk I will discuss the Mikhlin-H\"ormander multiplier theorem for $L^p$ boundedness of Fourier multipliers in which the multiplier belongs to a fractional Sobolev space with smoothness $s$. I will show that this theorem does not hold in the limiting case $|1/p - 1/2|=s/n$. I will also present a sharp variant of this theorem involving a space of Lorentz-Sobolev type. Some of the results presented in this talk were obtained in collaboration with Loukas Grafakos.

$L^p$ restriction of eigenfunctions to random Cantor-type sets

Series
Analysis Seminar
Time
Wednesday, September 26, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Suresh Eswarathasan – Cardiff University
Abstract: Let $(M,g)$ be a compact Riemannian n-manifold without boundary. Consider the corresponding $L^2$-normalized Laplace-Beltrami eigenfunctions. Eigenfunctions of this type arise in physics as modes of periodic vibration of drums and membranes. They also represent stationary states of a free quantum particle on a Riemannian manifold. In the first part of the lecture, I will give a survey of results which demonstrate how the geometry of $M$ affects the behaviour of these special functions, particularly their “size” which can be quantified by estimating $L^p$ norms. In joint work with Malabika Pramanik (U. British Columbia), I will present in the second part of my lecture a result on the $L^p$ restriction of these eigenfunctions to random Cantor-type subsets of $M$. This, in some sense, is complementary to the smooth submanifold $L^p$ restriction results of Burq-Gérard-Tzetkov ’06 (and later work of other authors). Our method includes concentration inequalities from probability theory in addition to the analysis of singular Fourier integral operators on fractals.

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