Sonstige Vorträge

Pro-etale cohomology of rigid-analytic spaces with Q_p-coefficients has some surprising features: it is not A^1-invariant and no general finiteness theorems over Q_p are true. It has been observed in recent years that these particularities can be explained by viewing the pro-etale cohomology as (quasi-)coherent cohomology on the Fargues-Fontaine curve. I want to explain joint work in progress with Arthur-Cesar Le Bras and Lucas Mann, which aims to fully implement this idea by developing a six functor formalism with values in solid quasi-coherent sheaves on relative Fargues-Fontaine curves.

The notion of global hyperbolicity was introduced by Jean Leray in 1952 to obtain global uniqueness of solutions to nonlinear wave equations. Globally hyperbolic spacetimes subsequently turned out to be the right geometric setting not only for the well-posedness of the initial value formulation for the Einstein equations in General Relativity but also for the singularity theorems of Penrose and Hawking and several splitting results in Lorentzian geometry. We review the rich history and omnipresence of global hyperbolicity in General Relativity. Then we present a surprising new characterization of globally hyperbolic spacetimes that makes use of ideas and tools from metric geometry.

*Abstract*: Rational ellipticity is a very strong condition on a topological space, which in particular forces it to have "simple topology''. Given its conjectured relation to manifolds with non-negative sectional curvature, a number of previous works has focused on finding geometric criteria that imply rational ellipticity. In this talk, I will describe a new criterion for a Riemannian G-manifold to be rationally elliptic, which generalizes most of the previously known ones. As an application, we will prove that non-negatively curved manifolds with an infinitesimally polar cohomogeneity 3 action must be rationally elliptic. This is joint work with Elahe Khalili Samani.