Vorträge des SFB 1442

Global G-shtukas relate to local G-shtukas as abelian varieties do to p-divisible groups. In particular, their moduli spaces are the function field analogues of Shimura varieties. Generalizations of the well-known Eichler-Shimura congruence relation have been proven for various Shimura varieties using a method introduced in the book of Faltings and Chai on degeneration of abelian varieties. In this talk, I will explain how to adapt this method to the function field setting to prove the ordinary part of the congruence relation for moduli spaces of global G-shtukas, and how the Hodge-Newton filtration fits into this picture.

Einstein's introduction of the cosmological constant to general relativity provided a mathematical framework for the study of the universe in the large. After a discussion of the early discoveries and ruminations of A Einstein and W de Sitter, I will move on to describe the Kerr de Sitter geometry which models a rotating black hole in an expanding universe. I will motivate the Cauchy problem in the context of the Einstein vacuum equations with positive cosmological constant, and present a recent resolution of the non-linear stability problem for the cosmological region. Among other works, the talk describes contributions by H Friedrich, P Hintz and A Vasy, and my recent joint work with G Fournodavlos.

Abstract: Homological stability is a pattern in the homology of families of spaces and groups. Examples of groups with homological stability include braid groups, symmetric groups, general linear groups, and mapping class groups. I will describe applications of this phenomenon to questions in analytic number theory. Specifically, I will report on joint work with Patzt, Petersen, and Randal-Williams on a stability pattern called uniform twisted homological stability and describe applications of these results to a conjecture of Conrey-Farmer-Keating-Rubinstein-Snaith on moments of quadratic L-functions over function fields.

There is an idea, going back to work of Krasner, that p-adic fields tend to function fields as absolute ramification tends to infinity. We will present a new way of rigorizing this idea, as well as give applications to the local Langlands correspondence of Fargues?Scholze.

Abstract: The Dehn invariant is known to many as the satisfying solution to Hilbert?s 3rd problem: a three-dimensional polyhedron P can be cut into pieces and reassembled into a polyhedron Q if and only if Q and P have not only the same volume, but also the same Dehn invariant. Generalised versions of Hilbert?s 3rd problem concern the so-called scissors congruence groups of euclidean, hyperbolic and spherical geometry in varying dimensions, and in these contexts one can define a generalised Dehn invariant. In the spherical case, Sah showed that the Dehn invariant makes the scissors congruence groups into a graded Hopf algebra. Zakharevich has shown that one can lift the scissors congruence group to a K-theory spectrum. In this talk I will discuss a lift of the Dehn invariant to the spectrum level, and we will see how it gives rise to a spectral version of Sah?s Hopf algebra. This talk is based on joint work in progress with Inbar Klang, Cary Malkiewich, David Mehrle and Thor Wittich.

Local models of Shimura varieties are projective flat schemes over the spectrum of a discrete valuation ring. The importance of local models lies in the fact that under some assumptions they model the singularities that arise in the reduction modulo p of Shimura varieties. In this talk, we will first introduce the notion of local models for some unitary and orthogonal Shimura varieties. Building on this, we will resolve the singularities of these models, leading to regular integral models for the corresponding Shimura varieties.