Vorträge des SFB 1442

Since Hawking proved his singularity theorem, it is clear that singularities of big bang type typically occur in cosmological solutions to Einstein?s equations. Moreover, due to the work of Belinskii, Khalatnikov and Lifschitz (BKL), there is a proposal concerning the nature of these singularities. For many matter models, solutions are expected to exhibit chaotic dynamics governed by the so-called BKL map, but for some matter models, the dynamics are expected to be convergent/quiescent. The purpose of the talk is to discuss quiescent singularities, with an emphasis on a geometric notion of initial data on the singularity. In particular, we present a general condition on initial data ensuring big bang formation, curvature blow up and solutions that induce data on the singularity

Topological Hochschild homology (THH) has been used successfully to construct prismatic cohomology, the most powerful cohomology theory that we currently have for formal schemes over Z_p. But for rigid-analytic varieties over Q_p, or even global objects like varieties over Q, THH is less useful: For rational inputs, THH will always vanish modulo p, and so it can't have any interesting comparisons to, say, étale cohomology with torsion coefficients. In this talk, I'll explain a refinement of THH (due to Efimov and Scholze) that overcomes these issues, and I'll show a few promising computations. I'll also sketch how one should be able to recover Scholze's analytic Habiro cohomology from refined THH.

Partial group actions on spaces by homeomorphisms between open sets provide a rich class of C*-algebras via the crossed product construction. We start an in-depth investigation when a general Hausdorff étale groupoid is a transformation groupoid by a partial action. Our negative results contain the first examples of Hausdorff étale groupoids which do not arise as transformation groupoids of partial group actions and which are not inner amenable in the sense of Anantharaman-Delaroche. On the positive side, we prove that many higher rank graph algebras as well as many unital UCT Kirchberg algebras can be realized via partial group actions. This is joint work with Alcides Buss.

A line field on a manifold is a smooth and continuous assignment of a tangent line to each point on the manifold. The projective span of a smooth manifold is the maximal number of linearly independent line fields. After motivating the study of this fascinating numerical invariant, I will not only give a necessary condition for the existence of linearly independent line fields on (almost) complex manifolds, but I will also explain that this condition is additionally sufficient in certain cases. Finally, we will apply these necessary and sufficient conditions to obtain a refinement of the Schwarzenberger condition that dictates which cohomology classes can be the Chern classes of a complex vector bundle (with prescribed line bundle splitting properties) over complex projective space. This is joint work with Nikola Sadovek (Dresden) based on [arXiv:2411.14161] (recently accepted for publication in Mathematische Annalen).