Oberseminare und sonstige Vorträge

The spinor flow of Ammann-Weiß-Witt on a compact manifolds $M$ is a geometric flow equation for pairs $(g,\phi)$ of a Riemannian metric $g$ and a unit spinor field $\phi$ whose critical points in dimensions greater or equal to 3 are Ricci-flat Riemannian manifolds $(M,g)$ of special holonomy together with a parallel spinor field $\phi$. In my talk, I will give an introduction to this flow in the general setting and show how one may define a homogeneous version on compact homogeneous manifolds, which may also be extended to non-compact homogeneous manifolds. Moreover, I will present some results on the long-time behaviour of the usual and a volume-normalized version of the homogeneous spinor flow on specific homogeneous manifolds.

We consider a new class of potentially exotic group C*-algebras $C^*(\text{PF}_{p}^*(G))$ for a locally compact group $G$, and its connection with the class of potentially exotic group C*-algebras $C^*_{L^p}(G)$ introduced by Brown and Guentner \cite{BG}. Surprisingly, these two classes of C*-algebras are intimately related. By exploiting this connection, we show $C^*_{L^p}(G)=C^*(\text{PF}_{p}^*(G))$ for $p\in (2,\infty)$, and the C*-algebras $C^*_{L^p}(G)$ are pairwise distinct for $p\in (2,\infty)$ when $G$ belongs to a large class of nonamenable groups possessing the Haagerup property and either the rapid decay property or Kunze-Stein phenomenon by characterizing the positive definite functions that extend to positive linear functionals of $C^*_{L^p}(G)$ and $C^*(\text{PF}_{p}^*(G))$. This greatly generalizes earlier results of Okayasu \cite{Okay} and Wiersma on the pairwise distinctness of $C^*_{L^p}(G)$ for $20$ (recall $A_\pi\subseteq B_\pi$). Here $A_\pi$ and $B_\pi$ are the closed span of coefficients of $\pi$ in the norm and $w^*$-topology of the Fourier-Stieljes algebra $B(G)=C^*(G)^*$, respectively. In particular, we have $A_\pi\subseteq B_\pi$.\\ This is based on a joint work with M. Wiersma.

Abstract: We consider solutions to Ricci flow defined on manifolds M for a time interval (0,T) whose Ricci curvature is bounded uniformly in time from below, and for which the norm of the full curvature tensor at time t is bounded by c/t for some fixed constant c>1 for all t in (0,T). From previous works, it is known that if the solution is complete for all times t>0, then there is a limit metric space (M,d_0), as time t approaches zero. We show : if there is a region V on which (V,d_0) is *smooth*, then the solution can be extended smoothly to time zero on V.