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Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

In
this talk, I will discuss some polynomials that are best approximants
(in some sense!) to reciprocals of functions in some analytic function
spaces of the unit disk. I will examine the extremal
problem of finding a zero of minimal modulus, and will show how that
extremal problem is related to the spectrum of a certain Jacobi matrix
and real orthogonal polynomials on the real line.

Series: Analysis Seminar

t-Haar multipliers are examples of Haar multipliers were the symbol depends both on the frequency variable (dyadic intervals) and on the space variable, akin to pseudo differential operators. They were introduced more than 20 years ago, the corresponding multiplier when $t=1$ appeared first in connection to the resolvent of the dyadic paraproduct, the cases $t=\pm 1/2$ is intimately connected to direct and reverse inequalities for the dyadic square function in $L^2$, the case $t=1/p$ naturally appears in the study of weighted inequalities in $L^p$. Much has happened in the theory of weighted inequalities in the last two decades, highlights are the resolution of the $A_2$ conjecture (now theorem) by Hyt\"onen in 2012 and the resolution of the two weight problem for the Hilbert transform by Lacey, Sawyer, Shen and Uriarte Tuero in 2014. Among the competing methods used to prove these results were Bellman functions, corona decompositions, and domination by sparse operators. The later method has gained a lot of traction and is being widely used in contexts beyond what it was originally conceived for in work of Lerner, several of these new applications have originated here at Gatech. In this talk I would like to tell you what I know about t-Haar multipliers (some work goes back to my PhD thesis and joint work with Nets Katz and with my former students Daewon Chung, Jean Moraes, and Oleksandra Beznosova), and what we ought to know in terms of sparse domination.

Series: Analysis Seminar

In this talk I will introduce a Tb Theorem that characterizes all Calderón-Zygmund operators that extend compactly on L^p(R^n) by means of testing functions as general
as possible.
In the classical theory for boundedness, the testing functions satisfy a non-degeneracy property called accretivity, which essentially implies the existence of a positive lower bound
for the absolute value of the averages of the testing functions over all dyadic cubes. However, in the setting of compact operators, due to their better properties, the hypothesis of accretivity can be relaxed to a large extend.
As a by-product, the results also describe those Calderón-Zygmund operators whose boundedness can be checked with non-accretive testing functions.

Series: Analysis Seminar

The bispectral problem concerns the construction and the classification
of operators possessing a symmetry between the space and spectral
variables. Different versions of this problem can be solved using
techniques from integrable systems, algebraic geometry, representation
theory, classical orthogonal polynomials, etc. I will review the problem
and some of these connections and then discuss new results related to
the generic quantum superintegrable system on the sphere.