Seminars and Colloquia by Series

Wednesday, September 19, 2018 - 13:55 , Location: Skiles 005 , Marcin Bownik , University of Oregon , Organizer: Shahaf Nitzan
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.
Wednesday, September 12, 2018 - 13:55 , Location: Skiles 005 , Galyna Livshyts , Georgia Institute of Technology , , Organizer: Galyna Livshyts
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. 
Wednesday, September 5, 2018 - 13:55 , Location: Skiles 006 , Ionel Popescu , Georgia Institute of Technology , Organizer: Galyna Livshyts
 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.  
Wednesday, August 29, 2018 - 01:55 , Location: Skiles 154 , Michael Lacey , Georgia Tech , Organizer: Michael Lacey
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. 
Wednesday, April 25, 2018 - 01:55 , Location: Skiles 005 , March Boedihardjo , UCLA , Organizer: Shahaf Nitzan
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
Wednesday, April 18, 2018 - 13:55 , Location: Skiles 005 , Benjamin Jaye , Clemson University , , Organizer: Galyna Livshyts
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
Wednesday, April 11, 2018 - 13:55 , Location: Skiles 005 , Kateryna Tatarko , University of Alberta , , Organizer: Galyna Livshyts
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.
Wednesday, March 28, 2018 - 13:55 , Location: Skiles 005 , Laura Cladek , UCLA , , Organizer: Michael Lacey
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.
Wednesday, March 14, 2018 - 13:55 , Location: Skiles 005 , Jose Conde Alonso , Brown University , , Organizer: Galyna Livshyts
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.
Wednesday, March 7, 2018 - 13:55 , Location: Skiles 005 , Amalia Culiuc , Georgia Tech , , Organizer: Galyna Livshyts
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.