# Joint IAS/PU Number Theory

## Extremal cases of Rapoport-Zink spaces

Michael Rapoport
Universit&auml;t Bonn, University of Maryland
October 10, 2019

This talk is about qualitative properties of the underlying scheme of Rapoport-Zink formal moduli spaces of p-divisible groups, resp. Shtukas. We single out those cases when the dimension of this underlying scheme is zero, resp. those where the dimension is maximal possible. The model case for the first alternative is the Lubin-Tate moduli space, and the model case for the second alternative is the Drinfeld moduli space. We exhibit a complete list in both cases.

## Golden Gates in PU(n) and the Density Hypothesis.

Shai Evra
Member, School of Mathematics
September 26, 2019
In their seminal work from the 80’s, Lubotzky, Phillips and Sarnak gave explicit constructions of topological generators for PU(2) with optimal covering properties. In this talk I will describe some recent works that extend the construction of LPS to higher rank compact Lie groups.