Vorträge in der Woche 05.05.2025 bis 11.05.2025

Dienstag, 06.05.2025: Tannakian categories

Giacomo Gavelli

Donnerstag, 08.05.2025: Lattice points in high-dimensional spheres and balls with small radii and dimension-free estimates for discrete maximal functions

Dr. Blazej Wróbel (University of Wroclaw)

Counting the number of integer lattice points on the surface of a $d$-dimensional sphere of radius $r$ is a theme that has been studied for several centuries. Still, there are questions which remain open to this day. In particular, asymptotic formulas which are uniform in both $d$ and $r$ are not known for all possible ranges of the dimensions and radii. I will talk about a recent result, together with my student J. Niksinski, in which we establish such an asymptotic formula within the range $r^2<0.1d.$ I will also describe applications of this asymptotic formula to dimension-free estimates of discrete maximal functions over spheres and balls. This makes progress on a question we studied several years ago with J. Bourgain, M. Mirek, and E. M. Stein.