Lengths of closed geodesics in manifolds of positive scalar curvature

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. Time permitting, I will describe other results about geometry and topology of 3-manifolds proved using minimal surfaces. This is a joint work with Davi Maximo and Regina Rotman.