Zoom link: https://princeton.zoom.us/j/594605776

I will discuss questions related to the volume of closed manifolds with bounded sectional curvature, in dimensions at least 3. For any small enough dimensional constant \epsilon>0, I will explain how to define a functional V_\epsilon and a procedure which, starting from a smooth closed manifold M, gives a Riemannian manifold with sectional curvature bounded by 1, with volume equal to V_\epsilon(M) and whose \epsilon-thick part has a volume minimizing property. This procedure gives the hyperbolic metric if M is hyperbolic. There are also estimates of V_\epsilon for complex surfaces. The functional V_\epsilon is closely related to the minimal volume first considered by Gromov.