Seminars and Colloquia by Series

$L^p$ 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.

Approximate similarity of operators on l^p

Series
Analysis Seminar
Time
Wednesday, April 25, 2018 - 01:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
March BoedihardjoUCLA
Abstract: I will state a version of Voiculescu's noncommutative Weyl-von Neumann theorem for operators on l^p that I obtained. This allows certain classical results concerning unitary equivalence of operators on l^2 to be generalized to operators on l^p if we relax unitary equivalence to similarity. For example, the unilateral shift on l^p, 1

On the probability that a stationary Gaussian process with spectral gap remains non-negative on a long interval

Series
Analysis Seminar
Time
Wednesday, April 18, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Benjamin JayeClemson University
We discuss the probability that a continuous stationary Gaussian process on whose spectral measure vanishes in a neighborhood of the origin stays non-negative on an interval of long interval. Joint work with Naomi Feldheim, Ohad Feldheim, Fedor Nazarov, and Shahaf Nitzan

An upper bound on the smallest singular value of a square random matrix

Series
Analysis Seminar
Time
Wednesday, April 11, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kateryna TatarkoUniversity of Alberta
Consider an n by n square matrix with i.i.d. zero mean unit variance entries. Rudelson and Vershynin showed that its smallest singular value is bounded from above by 1/sqrt{n} with high probability, under the assumption of the bounded fourth moment of the entries. We remove the assumption of the bounded fourth moment, thereby extending the result of Rudelson and Vershynin to a wide range of distributions.

Quantitative additive energy estimates for regular sets and connections to discretized sum-product theorems

Series
Analysis Seminar
Time
Wednesday, March 28, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
We prove new quantitative additive energy estimates for a large class of porous measures which include, for example, all Hausdorff measures of Ahlfors-David subsets of the real line of dimension strictly between 0 and 1. We are able to obtain improved quantitative results over existing additive energy bounds for Ahlfors-David sets by avoiding the use of inverse theorems in additive combinatorics and instead opting for a more direct approach which involves the use of concentration of measure inequalities. We discuss some connections with Bourgain's sum-product theorem.

Calder\'on-Zygmund operators cannot be bounded on $L^2$ with totally irregular measures

Series
Analysis Seminar
Time
Wednesday, March 14, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jose Conde AlonsoBrown University
We consider totally irregular measures $\mu$ in $\mathbb{R}^{n+1}$, that is, $$\limsup_{r\to0}\frac{\mu(B(x,r))}{(2r)^n} >0 \;\; \& \;\; \liminf_{r\to0}\frac{\mu(B(x,r))}{(2r)^n}=0$$for $\mu$ almost every $x$. We will show that if $T_\mu f(x)=\int K(x,y)\,f(y)\,d\mu(y)$ is an operator whose kernel $K(\cdot,\cdot)$ is the gradient of the fundamental solution for a uniformly elliptic operator in divergence form associated with a matrix with H\"older continuous coefficients, then $T_\mu$ is not bounded in $L^2(\mu)$.This extends a celebrated result proved previously by Eiderman, Nazarov and Volberg for the $n$-dimensional Riesz transform and is part of the program to clarify the connection between rectifiability of sets/measures on $\mathbb{R}^{n+1}$ and boundedness of singular integrals there. Based on joint work with Mihalis Mourgoglou and Xavier Tolsa.