Termine

I will talk about the mixing time which is the time it takes for a random walk to reach equilibrium. My focus will be on the cutoff phenomenon observed when the transition to equilibrium happens abruptly in time. I will survey the developments in the last 30 years and present a recent universality result for graphs with a random matching that was obtained in collaboration with J. Hermon and A. Sly.

Further information

We discuss a result from 2020 by Conant-Pillay on Approximate Subgroups with Bounded VC Dimension as well as more recent developments by, for example, Conant-Terry. The aim is to highlight the use of model theoretic techniques to motivate and/or prove results in arithmetic combinatorics, particularly approximate group theory.

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.