Seminars and Colloquia by Series

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
Laura CladekUCLA
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.

Matrix weights and the A2 conjecture

Series
Analysis Seminar
Time
Wednesday, March 7, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Amalia CuliucGeorgia Tech
An overarching problem in matrix weighted theory is the so-called A2 conjecture, namely the question of whether the norm of a Calderón-Zygmund operator acting on a matrix weighted L2 space depends linearly on the A2 characteristic of the weight. In this talk, I will discuss the history of this problem and provide a survey of recent results with an emphasis on the challenges that arise within the setup.

A sharp Schroedinger maximal estimate in two dimensions

Series
Analysis Seminar
Time
Wednesday, February 28, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Xiumin DuInstitute for Advanced Study
Joint with Guth and Li, recently we showed that the solution to the free Schroedinger equation converges to its initial data almost everywhere, provided that the initial data is in the Sobolev space H^s(R^2) with s>1/3. This is sharp up to the endpoint, due to a counterexample by Bourgain. This pointwise convergence problem can be approached by estimates of Schroedinger maximal functions, which have some similar flavor as the Fourier restriction estimates. In this talk, I'll first show how to reduce the original problem in three dimensions to an essentially two dimensional one, via polynomial partitioning method. Then we'll see that the reduced problem asks how to control the size of the solution on a sparse and spread-out set, and it can be solved by refined Strichartz estimates derived from l^2 decoupling theorem and induction on scales.

Hamming cube and martingales: duality

Series
Analysis Seminar
Time
Wednesday, February 21, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Paata IvanisviliPrinceton University
I will speak how to ``dualize'' certain martingale estimates related to the dyadic square function to obtain estimates on the Hamming and vice versa. As an application of this duality approach, I will illustrate how to dualize an estimate of Davis to improve a result of Naor--Schechtman on the real line. If time allows we will consider one more example where an improvement of Beckner's estimate will be given.

Finite Balian Low Theorems in $\mathbb{R}^d$

Series
Analysis Seminar
Time
Wednesday, February 14, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Josiah ParkGeorgia Institute of technology
We study Balian-Low type theorems for finite signals in $\mathbb{R}^d$, $d\geq 2$.Our results are generalizations of S. Nitzan and J.-F. Olsen's recent work and show that a quantity closelyrelated to the Balian-Low Theorem has the same asymptotic growth rate, $O(\log{N})$ for each dimension $d$. Joint work with Michael Northington.

A constrained optimization problem for the Fourier transform

Series
Analysis Seminar
Time
Wednesday, February 7, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dominique Maldague UC Berkeley
Among functions $f$ majorized by indicator functions $1_E$, which functions have maximal ratio $\|\widehat{f}\|_q/|E|^{1/p}$? I will briefly describe how to establish the existence of such functions via a precompactness argument for maximizing sequences. Then for exponents $q\in(3,\infty)$ sufficiently close to even integers, we identify the maximizers and prove a quantitative stability theorem.

On the role of symmetry in geometric inequalities

Series
Analysis Seminar
Time
Wednesday, January 31, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Galyna LivshytsGeorgia Tech
In the recent years, a number of conjectures has appeared, concerning the improvement of the inequalities of Brunn-Minkowski type under the additional assumptions of symmetry; this includes the B-conjecture, the Gardner-Zvavitch conjecture of 2008, the Log-Brunn-Minkowski conjecture of 2012, and some variants. The conjecture of Gardner and Zvavitch, also known as dimensional Brunn-Minkowski conjecture, states that even log-concave measures in R^n are in fact 1/n-concave with respect to the addition of symmetric convex sets. In this talk we shall establish the validity of the Gardner-Zvavitch conjecture asymptotically, and prove that the standard Gaussian measure enjoys 0.3/n concavity with respect to centered convex sets. Some improvements to the case of general log-concave measures shall be discussed as well. This is a joint work with A. Kolesnikov.

Host-Kra norms and Gowers structure on Euclidean spaces

Series
Analysis Seminar
Time
Wednesday, January 24, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Martina NeumanUC Berkeley
The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.

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