Georgia Institute of Technology College of Sciences

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.

In the beginning, the basics about random matrix models and some facts about normal random matrices in relation with conformal map- pings will be explained. In the main part we will show that for Gaussian random normal matrices the eigenvalues will fill an elliptically shaped do- main with constant density when the dimension n of the matrices tends to infinity. We will sketch a proof of universality, which is based on orthogonal polynomials and an identity which plays a similar role as the Christoffel- Darboux formula in Hermitian random matrices.

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.

In joint work with J. Martinez-Garcia we study the classification problem of asymptotically log del Pezzo surfaces in algebraic geometry. This turns out to be equivalent to understanding when certain convex bodies in high-dimensions intersect the cube non-trivially. Beyond its intrinsic interest in algebraic geometry this classification is relevant to differential geometery and existence of new canonical metricsin dimension 4.

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.

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

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.

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).

We will discuss several open problems concerning unique determination of convex bodies in the n-dimensional Euclidean space given some information about their projections or sectionson all sub-spaces of dimension n-1. We will also present some related results.

In this talk we will discuss some some extremal problems for polynomials. Applications to the problems in discrete dynamical systems as well as in the geometric complex analysis will be suggested.