**Date & Time:** Wednesday, September 10, 2014, 16:00-17:00.

**Venue:** Ramanujan Hall

**Speaker:** Samik Basu, RKM Vivekananda University

**Title:** Homotopy Groups and Closed Geodesics on 4-manifolds

**Abstract:** A simply connected, closed four manifold, is associated to a simply connected, closed, spin five manifold which occurs as the total space of a circle bundle. This leads to several consequences: the homotopy groups of such a four manifold are determined by its second Betti number, and the ranks of the homotopy groups can be explicitly calculated. We also show that for a generic metric on such a smooth four manifold with second Betti number at least three the number of geometrically distinct periodic geodesics of length at most l grows exponentially as a function of l.