Vorträge in der Woche 05.05.2025 bis 11.05.2025

Dienstag, 06.05.2025: Tannakian categories

Giacomo Gavelli

Mittwoch, 07.05.2025: 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.

Donnerstag, 08.05.2025: A generalization of the ADM mass for asymptotically Euclidean manifolds of weak regularity

Benjamin Meco (Uppsala University)

Recently, Gicquaud and Sakovich introduced a new definition of mass for weakly regular asymptotically hyperbolic manifolds. Inspired by their work, in collaboration with Stig Lundgren we propose a new definition of mass for weakly regular asymptotically Euclidean manifolds. After a preliminary discussion of the classical ADM mass, we present our new definition and describe the class of metrics to which it applies. We then investigate the conditions under which this mass is well-defined, coincides with the classical ADM mass, and remains coordinate invariant. Finally, we offer an alternative formulation of the mass in terms of the Einstein tensor.