Decidability and periodicity of translational tilings

Speaker: 

Rachel Greenfeld

Institution: 

UCLA

Time: 

Thursday, February 24, 2022 - 11:00am

Host: 

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

Speaker: 

Jose Madrid

Institution: 

UCLA

Time: 

Thursday, February 17, 2022 - 11:00am

Host: 

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

Speaker: 

Gabriel Rivière

Institution: 

Université de Nantes

Time: 

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

Host: 

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.

On Maz’ya’s \Phi-inequalities.

Speaker: 

Dmitriy Stolyarov

Institution: 

St. Petersburg State University

Time: 

Thursday, November 4, 2021 - 11:00am

Location: 

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

I will speak about an unusual way to correct the (invalid) endpoint case of the Hardy—Littlewood—Sobolev inequality. Usually the correction is done by imposing additional linear constraints on the function we apply the Riesz potential to. Being the gradient of another function is an example of such a constraint. The inequalities obtained this way are often called Bourgain—Brezis inequalities. In 2010, Maz’ya suggested another approach: instead of constraining the right hand side we should replace the L_p norm on the left with an expression \Phi, which alongside with having the same homogeneity properties as the L_p norm, possesses additional cancellations. He conjectured that if \Phi satisfies a natural necessary condition, then the modified Hardy—Littlewood—Sobolev inequality holds true. I will try to survey the proof of Maz’ya’s conjecture. Based on https://arxiv.org/abs/2109.08014

Estimating average of function on Boolean cube without risk

Speaker: 

March Boedihardjo

Institution: 

UCI

Time: 

Thursday, October 7, 2021 - 11:00am

Location: 

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

Strong law of large numbers gives a method to estimate the
average of a function on the Boolean cube so that it is accurate with
high probability. But there is still a little risk that it is
inaccurate. I will present a polynomial time method to estimate the
averages of certain functions on Boolean cube without risk of being
inaccurate.

A moment ratio bound for polynomials on the boolean cube

Speaker: 

Alex Samorodnitsky

Institution: 

The Hebrew University of Jerusalem

Time: 

Thursday, November 18, 2021 - 11:00am

Location: 

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

We prove an almost tight upper bound on the ratio ||f||_p / ||f||_2, when f is a polynomial of a given degree on the Boolean cube {0,1}^n and describe some applications.  In particular, we describe a family of hypercontractive inequalities for functions on {0,1}^n which take into account the concentration of a function.

CANCELED

Speaker: 

Oscar Dominguez [CANCELED]

Institution: 

Universidad Complutense de Madrid, Spain

Time: 

Friday, April 10, 2020 - 3:00am

Host: 

Location: 

RH 340P

Maximal functions play a central role in the study of differentiation,
singular integrals and almost everywhere convergence. With Sergey Tikhonov
(ICREA, Barcelona) we recently proved some pointwise estimates for maximal
functions in terms of smoothness and rearrangements. I plan to discuss the recent
progress on these topics and some applications. In particular, I will discuss the
Fefferman-Stein inequality for the sharp maximal function for r.i. spaces which
are close to L∞.

The multilinear restriction estimates for hypersurfaces with curvature

Speaker: 

Ioan Bejenaru

Institution: 

UCSD

Time: 

Wednesday, February 19, 2020 - 3:00pm to 3:50pm

Host: 

Location: 

340P

Abstract: I will present an overview of the multilinear restriction theory, with an emphasis on the case when the hypersurfaces have some curvature. I will discuss a new result: the case of n-1 hypersurfaces in n dimensions where a fairly general theory is developed. 

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