Diffeomorphisms of discs and positive scalar curvature

Abstract: Let $\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(D^k)$ be the space of diffeomorphisms of the $k$-disc fixing the boundary pointwise. Understanding the topology of $\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(D^k)$ is a longstanding and difficult problem in differential topology. In this talk I will report on recent work with Thomas Schick and Wolfgang Steimle which shows that for $k>5$ the homotopy groups ${\pi }_{\ast }\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(D^k)$ have non-zero $8$-periodic $2$-torsion detected in real K-theory. I will then discuss applications to the space of positive scalar curvature metrics on spin manifolds of dimension $6$ or greater.