Abstract: Let \({\rm Diff}(D^k)\) be the space of diffeomorphisms of the \(k\)-disc fixing the boundary pointwise. Understanding the topology of \({\rm Diff}(D^k)\) 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 \(\pi_* {\rm Diff}(D^k)\) 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.