Seminars and Colloquia by Series

Integral geometric regularity

Series
Analysis Seminar
Time
Wednesday, October 31, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joe FuUGA
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 RyaboginKent 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 HuangVanderbilt 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 SlavikovaUniversity 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 EswarathasanCardiff 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.

Exponential frames and syndetic Riesz sequences

Series
Analysis Seminar
Time
Wednesday, September 19, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Marcin BownikUniversity of Oregon
In this talk we shall explore some of the consequences of the solution to the Kadison-Singer problem. In the first part of the talk we present results from a joint work with Itay Londner. We show that every subset $S$ of the torus of positive Lebesgue measure admits a Riesz sequence of exponentials $\{ e^{i\lambda x}\} _{\lambda \in \Lambda}$ in $L^2(S)$ such that $\Lambda\subset\mathbb{Z}$ is a set with gaps between consecutive elements bounded by $C/|S|$. In the second part of the talk we shall explore a higher rank extension of the main result of Marcus, Spielman, and Srivastava, which was used in the solution of the Kadison-Singer problem.

On the Koldobsky's slicing conjecture for measures

Series
Analysis Seminar
Time
Wednesday, September 12, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Galyna LivshytsGeorgia Institute of Technology
Koldobsky showed that for an arbitrary measure on R^n, the measure of the largest section of a symmetric convex body can be estimated from below by 1/sqrt{n}, in with the appropriate scaling. He conjectured that a much better result must hold, however it was recemtly shown by Koldobsky and Klartag that 1/sqrt{n} is best possible, up to a logarithmic error. In this talk we will discuss how to remove the said logarithmic error and obtain the sharp estimate from below for Koldobsky's slicing problem. The method shall be based on a "random rounding" method of discretizing the unit sphere. Further, this method may be effectively applied to estimating the smallest singular value of random matrices under minimal assumptions; a brief outline shall be mentioned (but most of it shall be saved for another talk). This is a joint work with Bo'az Klartag.

Free probability inequalities on the circle and a conjecture

Series
Analysis Seminar
Time
Wednesday, September 5, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ionel PopescuGeorgia Institute of Technology
I will discuss some free probability inequalities on the circle which can be seen in two different ways, one is via random matrix approximation, and another one by itself. I will show what I believe to be the key of these new forms, namely the fact that the circle acts on itself. For instance the Poincare inequality has a certain form which reflects this aspect. I will also briefly show how a transportation inequality can be discussed and how the standard Wasserstein distance can be modified to introduce this interesting phenomena. I will end the talk with a conjecture and some supporting evidence in the classical world of functional inequalities.

Sparse bounds for Spherical Averages

Series
Analysis Seminar
Time
Wednesday, August 29, 2018 - 01:55 for 1 hour (actually 50 minutes)
Location
Skiles 154
Speaker
Michael LaceyGeorgia Tech
Spherical averages, in the continuous and discrete setting, are a canonical example of averages over lower dimensional varieties. We demonstrate here a new approach to proving the sparse bounds for these opertators. This approach is a modification of an old technique of Bourgain.

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