I will prove Gromov's conjecture that every 3-manifold of positive scalar curvature contains a short closed geodesic. The proof uses Min-Max theory of minimal surfaces and a combinatorial version of mean curvature flow. This is a joint work with Davi Maximo and Regina Rotman.

Einstein metrics with non-positive scalar curvature are conjectured to be linearly stable. On spin manifolds, the existence of parallel spinors provides a spinorial criterion for stability. In this talk I explain how this mechanism extends to non-spin manifolds using twisted spinors. In particular, we show that Einstein manifolds carrying parallel twisted pure spinors are linearly semistable. This yields stability results for a large class of spaces, including quaternionic Kähler manifolds of dimension greater than 8.

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