## Seminars and Colloquia by Series

### Sparse bounds for discrete spherical maximal functions

Series
Analysis Seminar
Time
Wednesday, February 6, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dario Alberto MenaUniversity of Costa Rica
We prove sparse bounds for the spherical maximal operator of Magyar,Stein and Wainger. The bounds are conjecturally sharp, and contain an endpoint esti-mate. The new method of proof is inspired by ones by Bourgain and Ionescu, is veryefficient, and has not been used in the proof of sparse bounds before. The Hardy-Littlewood Circle method is used to decompose the multiplier into major and minor arccomponents. The efficiency arises as one only needs a single estimate on each elementof the decomposition.

### Distance sets, lattice points, and decoupling estimates

Series
Analysis Seminar
Time
Wednesday, January 30, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alex IosevichUniversity of Rochester
We are going to discuss some recent results pertaining to the Falconer distance conjecture, including the joint paper with Guth, Ou and Wang establishing the $\frac{5}{4}$ threshold in the plane. We are also going to discuss the extent to which the sharpness of our method and similar results is tied to the distribution of lattice points on convex curves and surfaces.

### Valuations on convex sets and integral geometry

Series
Analysis Seminar
Time
Wednesday, January 23, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Semyon AleskerTel Aviv University
Valuations are finitely additive measures on convex compact subsets of a finite dimensional vector space. The theory of valuations originates in convex geometry. Valuations continuous in the Hausdorff metric play a special role, and we will concentrate in the talk on this class of valuations. In recent years there was a considerable progress in the theory and its applications. We will describe some of the progress with particular focus on the multiplicative structure on valuations and its applications to kinematic formulas of integral geometry.

### Sparse domination and the strong maximal function

Series
Analysis Seminar
Time
Wednesday, January 16, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alexander BarronBrown University
There has been recent interest in sparse bounds for various operators that arise in harmonic analysis. Perhaps the most basic "sparse" result is a pointwise bound for the dyadic Hardy-Littlewood maximal function. It turns out that the direct analogue of this result does not hold if one adds an extra dilation parameter: the dyadic strong maximal function does not admit a pointwise sparse bound or a sparse bound involving L^1 forms (both of which hold in the one-parameter setting). The proof is based on the construction of a certain pair of extremal point sets. This is joint work with Jose Conde-Alonso, Yumeng Ou, and Guillermo Rey.

### Fuglede's spectral-set conjecture.

Series
Analysis Seminar
Time
Wednesday, December 5, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rachel GreenfeldBar Ilan University
A set $\Omega\subset \mathbb{R}^d$ is called spectral if the space $L^2(\Omega)$ admits an orthogonal basis of exponential functions. Back in 1974 B. Fuglede conjectured that spectral sets could be characterized geometrically by their ability to tile the space by translations. Although since then the subject has been extensively studied, the precise connection between spectrality and tiling is still a mystery.>In the talk I will survey the subject and discuss some recent results, joint with Nir Lev, where we focus on the conjecture for convex polytopes.

### The fractal uncertainty principle

Series
Analysis Seminar
Time
Wednesday, November 28, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rui HanGeorgia Tech
Recently Bourgain and Dyatlov proved a fractal uncertainty principle (FUP), which roughly speaking says a function in $L^2(\mathbb{R})$ and its Fourier transform can not be simultaneously localized in $\delta$-dimensional fractal sets, $0<\delta<1$. In this talk, I will discuss a joint work with Schlag, where we obtained a higher dimensional version of the FUP. Our method combines the original approach by Bourgain and Dyatlov, in the more quantitative rendition by Jin and Zhang, with Cantan set techniques.

### Cotlar’s identity for Hilbert transforms---old and new stories.

Series
Analysis Seminar
Time
Wednesday, November 14, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tao MeiBaylor University
Cotlar’s identity provides an easy (maybe the easiest) argument for the Lp boundedness of Hilbert transforms. E. Ricard and I discovered a more flexible version of this identity, in the recent study of the boundedness of Hilbert transforms on the free groups. In this talk, I will try to introduce this version of Cotlar’s identity and the Lp Fourier multipliers on free groups.

### Portraits of RIFs: their singularities and unimodular level sets on T^2

Series
Analysis Seminar
Time
Wednesday, November 7, 2018 - 10:14 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kelly BickelBucknell University
This talk concerns two-variable rational inner functions phi with singularities on the two-torus T^2, the notion of contact order (and related quantities), and its various uses. Intuitively, contact order is the rate at which phi’s zero set approaches T^2 along a coordinate direction, but it can also be defined via phi's well-behaved unimodular level sets. Quantities like contact order are important because they encode information about the numerical stability of phi, for example when it belongs to Dirichlet-type spaces and when its partial derivatives belong to Hardy spaces. The unimodular set definition is also useful because it allows one to “see” contact order and in some sense, deduce numerical stability from pictures. This is joint work with James Pascoe and Alan Sola.

### Integral geometric regularity

Series
Analysis Seminar
Time
Wednesday, October 31, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joe FuUGA
The centerpiece of the subject of integral geometry, as conceived originally by Blaschke in the 1930s, is the principal kinematic formula (PKF). In rough terms, this expresses the average Euler characteristic of two objects A, B in general position in Euclidean space in terms of their individual curvature integrals. One of the interesting features of the PKF is that it makes sense even if A and B are not smooth enough to admit curvatures in the classical sense. I will describe the state of our understanding of the regularity needed to make it all work, and state some conjectures that would extend it.

### On the fifth Busemann-Petty problem

Series
Analysis Seminar
Time
Wednesday, October 24, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dmirty RyaboginKent State University
In 1956, Busemann and Petty posed a series of questions about symmetric convex bodies, of which only the first one has been solved.Their fifth problem asks the following.Let K be an origin symmetric convex body in the n-dimensional Euclidean space and let H_x be a hyperplane passing through the origin orthogonal to a unit direction x. Consider a hyperplane G parallel to H_x and supporting to K and let C(K,x)=vol(K\cap H_x)dist (0, G). (proportional to the volume of the cone spanned by the secion and the support point). If there exists a constant C such that for all directions x we have C(K,x)=C, does it follow that K is an ellipsoid?We give an affirmative answer to this problem for bodies sufficiently close to the Euclidean ball in the Banach Mazur distance.This is a joint work with Maria Alfonseca, Fedor Nazarov and Vlad Yaskin.