Continuous analytic capacity, rectifiability and holomorphic motions

Malik Younsi

Institution:

University of Hawaii

Time:

Thursday, June 30, 2022 - 3:00pm

Location:

RH 306

In this talk, I will present several geometric and analytic characterizations of purely unrectifiable quasicircles. These necessary and sufficient conditions are expressed in terms of various notions such as Dirichlet algebras, harmonic measure, analytic capacity and continuous analytic capacity. As an application, I will explain how to construct a compact set whose continuous analytic capacity does not vary continuously under a certain holomorphic motion. This answers a question raised by Paul Gauthier.

A stationary set method for estimating oscillatory integrals

Ruixiang Zhang

UC Berkeley

Time:

Thursday, May 19, 2022 - 11:00am

Location:

RH 306

Given a polynomial $P$ of constant degree in $d$ variables and consider the oscillatory integral $$I_P = \int_{[0,1]^d} e(P(\xi)) \mathrm{d}\xi.$$ Assuming $d$ is also fixed, what is a good upper bound of $|I_P|$? In this talk, I will introduce a stationary set'' method that gives an upper bound with simple geometric meaning. The proof of this bound mainly relies on the theory of o-minimal structures. As an application of our bound, we obtain the sharp convergence exponent in the two dimensional Tarry's problem for every degree via additional analysis on stationary sets. Consequently, we also prove the sharp $L^{\infty} \to L^p$ Fourier extension estimates for every two dimensional Parsell-Vinogradov surface whenever the endpoint of the exponent $p$ is even. This is joint work with Saugata Basu, Shaoming Guo and Pavel Zorin-Kranich.

The characteristic polynomial of sums of random permutations

Yizhe Zhu

UCI

Time:

Thursday, May 26, 2022 - 11:00am

Location:

RH 306

Let $A_n$ be the sum of $d$ permutations matrices of size $n×n$, each drawn uniformly at random and independently. We prove that $\det( I_n−zA_n/\sqrt{d})$ converges when $n\to\infty$ towards a random analytic function on the unit disk. As an application, we obtain an elementary proof of the spectral gap of random regular digraphs with a sharp constant. Our results are valid both in the regime where $d$ is fixed and for $d$ slowly growing with $n$. Joint work with Simon Coste and Gaultier Lambert.

Sharp multiplicative inequality on BMO

Pavel Zatitskiy

Institution:

Steklov Institute of Mathematics at St. Petersburg

Time:

Friday, February 25, 2022 - 11:00am

Location:

340P

We find the sharp constant C in the inequality $\|\phi\|_{L^r} \leq C \|\phi\|_{L^p}^{p/r} \|\phi\|_{BMO}^{1-p/r}$, where $1\leq p\leq r<+\infty$. We use the Bellman function machinery to solve this problem. The Bellman function of three variables corresponding to this problem has a rather complicated structure, however, we managed to provide the explicit formulas for this function. Based on joint works with D. Stolyarov, V. Vasyunin,  and  I. Zlotnikov.

Measures with given marginals and information economics

Speaker:

Fedor Sandomirskiy

Caltech

Time:

Thursday, March 3, 2022 - 11:00am

Location:

RH 306

There is a connection between the problem of Bayesian persuasion (an informed agent aims to induce desirable behavior of uninformed ones by tailoring the information available to them) and feasibility questions for measures with given marginals. We will discuss the two problems, their connection, and related open questions.

The talk is based on two papers:

"Feasible Joint Posterior Beliefs" with Itai Arieli, Yakov Babichenko, and Omer Tamuz

and "Private Private information" with Kevin He and Omer Tamuz.

Decidability and periodicity of translational tilings

Rachel Greenfeld

UCLA

Time:

Thursday, February 24, 2022 - 11:00am

Location:

Zoom ID: 954 8208 3189. Passcode: the last 4 digits of the zoom ID in the reverse order.

Translational tiling is a covering of a space using translated copies of some building blocks, called the “tiles”, without any positive measure overlaps.  Which are the possible ways that a space can be tiled?  In the talk, we will discuss the study of this question as well as its applications, and report on recent progress, joint with Terence Tao.

On classical inequalities for autocorrelations and autoconvolutions

UCLA

Time:

Thursday, February 17, 2022 - 11:00am

Location:

Zoom ID: 95482083189. passcode: the last 4 digits of zoom ID in the reverse order

We will discuss some convolution inequalities on the real line, the study of these problems is motivated by a classical problem in additive combinatorics about estimating the size of Sidon sets. We will also discuss many related open problems. This talk will be accessible for a broad audience.

Meromorphic continuation of Poincaré series

Gabriel Rivière

Institution:

Université de Nantes

Time:

Thursday, December 2, 2021 - 11:00am to 11:50am

Location:

Zoom ID: 949 5980 5461, Password: the last four digits of ID in the reverse order

Poincaré series are natural functions that arise in Riemannian geometry
when one wants to count the number of geodesic arcs of length less than
T between two given points on a compact manifold. I will begin with an
introduction on this topic. Then I will discuss some recent results with
N.V. Dang (Univ. Paris Sorbonne) showing that, in the case of negatively
curved manifolds, these series have a meromorphic continuation to the
whole complex plane. This can be shown by relating Poincaré series with
the resolvent of the geodesic vector field and by exploiting recent
results on this resolvent obtained through microlocal methods. If time
permits, I will also explain how the genus of a surface can be recovered
from the analysis of these series.

A Fredholm approach to scattering

Speaker:

Jesse Gell-Redman

Institution:

University of Melbourne

Time:

Thursday, October 28, 2021 - 11:00am

Location:

Zoom ID: 949 5980 546, Password: the last four digits of ID in the reverse order

We will give a friendly introduction to the scattering matrix for Schrodinger operators, and discuss how a new functional analytic approach to analysis of non-elliptic equations, due to Vasy, gives a conceptually attractive method for proving detailed regularity results for nonlinear scattering.  This is joint work with several groups of authors including Andrew Hassell, Sean Gomes, Jacob Shapiro, and Junyong Zhang.

The Nicest Average and New Uncertainty Principles for the Fourier Transform

Speaker:

Stefan Steinerberger

Institution:

University of Washington

Time:

Thursday, October 21, 2021 - 11:00am

Location:

zoom ID: 949 5980 5461. Password: the last four digits of the zoom ID in the reverse order

Two years ago, a colleague from economics asked me for the ”best” way to compute the average income over the last year. At first I didn’t understand but then he explained it to me: suppose you are given a real-valued function f(x) and want to compute a local average at a certain scale. What we usually do is to pick a nice probability measure u, centered at 0 and having standard deviation at the desired scale, and convolve f ∗ u. Classical candidates for u are the characteristic function or the Gaussian. This got me interested in finding the ”best” function u – this problem comes in two parts: (1) describing what one considers to be desirable properties of the convolution f ∗ u and (2) understanding which functions u satisfy these properties. I tried a basic notion for the first part, ”the convolution should be as smooth as the scale allows”, and ran into lots of really funky classical Fourier Analysis that seems to be new: (a) new uncertainty principles for the Fourier transform, (b) that potentially have the characteristic function as an extremizer, (c) which leads to strange new patterns in hypergeometric functions and (d) produces curious local stability inequalities. Noah Kravitz and I managed to solve two specific instances on the discrete lattice completely, this results in some sharp weighted estimates for polynomials on the unit interval – both the Dirichlet and the Fejer kernel make an appearance. The entire talk will be completely classical Harmonic Analysis, there are lots and lots of open problems and I will discuss several.