Seminars and Colloquia by Series

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.

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
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.