Idisplays

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.

Get an insight into the research of five new postdoctoral researchers of Mathematics Münster. In short scientific presentations they will introduce their topics.
After the talks, there will be the opportunity to exchange ideas while enjoying tea, coffee and cake in the Common Room.

• Catrin Mair, Homotopy theory in a condensed world
• Stefan Schrott, Optimal transport of stochastic processes
• Mathias Sonnleitner, Connecting and separating dots
• Ferdinand Wagner, Habiro Cohomology & Refined THH
• Alexander Van Werde, On the spectral determinacy of random graphs

In this talk, I will show an approach to deriving a priori bounds for coercive SPDEs based on scaling. The basic idea is to first show bounds for the equation with a small noise and then rescale the bounds to a global scale. While many equations that the approach can handle have been treated recently with other methods, its advantages are that it is quite simple and allows one to state a single result that is applicable to a variety of equations, such as rough differential equations and parabolic/elliptic SPDEs. Based on joint work with Massimiliano Gubinelli.

Symmetric monoidal categories are a fundamental object in many areas of mathematics, and among those, *rigid* categories - that is, those where every object admits a dual - are particularly well behaved. In this talk, after discussing some background, I will give a description of free rigid categories and describe some applications to K-theory, and specifically to chromatic redshift for rigid categories.

A smooth curve $\gamma: [0,1] \to \Ss^2$ is locally convex if its geodesic curvature is positive at every point. J.~A.~Little showed that the space of all locally positive curves $\gamma$ with $\gamma(0) = \gamma(1) = e_1$ and $\gamma'(0) = \gamma'(1) = e_2$ has three connected components $\cL_{-1,c}$, $\cL_{+1}$, $\cL_{-1,n}$. These spaces and variants have been discussed, among others, by B.~Shapiro, M.~Shapiro and B.~Khesin. Our first aim is to describe the homotopy type of these spaces. The connected component $\cL_{-1,c}$ is known to be contractible. We construct maps from $\cL_{+1}$ and $\cL_{-1,n}$ to $\Omega\Ss^3 \vee \Ss^2 \vee \Ss^6 \vee \Ss^{10} \vee \cdots$ and $\Omega\Ss^3 \vee \Ss^4 \vee \Ss^8 \vee \Ss^{12} \vee \cdots$, respectively, and show that they are (weak) homotopy equivalences. More generally, a smooth curve $\gamma: [0,1] \to \Ss^n \subset \RR^{n+1}$ is locally convex if $\det(\gamma(t),\ldots,\gamma^{n}(t)) > 0$ (for all $t$). We would like to understand the homotopy type of the space $\cL$ of locally convex curves with $\gamma^{(j)}(0) = \gamma^{(j)}(1) = e_{j+1}$ (for all $j \ne n$). We describe a CW complex with the same homotopy type. The homotopy type of $\cL$ is described for $n = 3$.

The modern theory of simple tracial C*-algebras is characterized by a rich collection of divisibility conditions, coming in numerous flavours and levels of strength. Z-stability and almost divisibility are primary examples of what might be called 'Cuntz-type' divisibility, while in the tracial setting we have counterparts like uniform property Gamma and tracial almost divisibility. On the dynamical side, analogous phenomena emerge through conditions like the small boundary property, the uniform Rokhlin property, and almost finiteness (in measure), often translating into 'relative' versions of the aforementioned properties for Cartan pairs. In this talk I will survey several of these regularity properties, alongside more recent notions introduced by Elliott and Niu, and explore some of the relationships between them. I will in furthermore try to relate some well-known instances of 'automatic centrality' which arises both in the setting of C*-algebras and in topological dynamics, by which I mean results that, under strong nuclearity assumptions, allow to upgrade non-central tracial divisibility conditions (e.g. tracial almost divisibility, small boundary property) to approximately central ones (uniform property Gamma, almost finiteness in measure).