The Heilbronn triangle problem is a classical problem in discrete geometry with several old and new close connections to various topics in extremal and additive combinatorics, graph theory, incidence geometry, harmonic analysis, and number theory. In this talk, we will survey a few of these stories, and discuss some recent developments. Based on joint works with Alex Cohen and Dmitrii Zakharov. 

In 1961, Grunbaum asked whether the centroid c(K) of a convex body K is the centroid of at least n + 1 different (n − 1)-dimensional sections of K through c(K). A few years later, Lowner asked to find the minimum number of hyperplane section of K passing through c(K) whose centroid is the same as c(K).

We give an answer to these questions for n ≥ 5. In particular, we construct a convex body which has only one section whose centroid coincides with the centroid of the body by using Fourier analytic tools and exploiting the existence of non-intersection bodies in these dimensions. Joint work with S. Myroshnychenko and V. Yaskin.

In this talk, I will discuss various bounds for the $L^p$ distance of polynomials on discrete hypercubes from Walsh tail spaces, extending some of Oleszkiewicz’s results (2017) for Rademacher sums. This is based on joint work with Alexandros Eskenazis (CNRS, Sorbonne University).

For planar billiard tables, the marked length spectrum encodes the lengths
of action (minus the length) minimizing orbits of a given rational rotation
number. For strictly convex tables, a renormalization of these lengths extends
to a continuous function called Mather’s beta function or the mean minimal
action. We show that using the algebraic structure of its Taylor coefficients,
one can prove C infinity compactness of marked length isospectral sets. This
gives a dynamical counterpart to the Laplace spectral results of Melrose,
Osgood, Phillips and Sarnak.

Abstract: In this talk, we discuss the semiclassical asymptotics for Bergman kernels in exponentially weighted spaces of holomorphic functions. We will first review a direct approach to the construction of asymptotic Bergman projections, developed by Deleporte--Hitrik--Sjöstrand in the case of real analytic weights, and Hitrik--Stone in the case of smooth weights. We shall then explore the case of Gevrey weights, which can be thought of as the interpolating case between the real analytic and smooth weights. In the case of Gevrey weights, we show that Bergman kernel can be approximated in certain Gevrey symbol class up to a Gevrey type small error, in the semiclassical limit. We will also introduce some microlocal analysis tools in the Gevrey setting, including Borel's lemma for symbols and complex stationary phase lemma. This talk is based on joint work with Hang Xu.