Title: Cones and symmetries of projective varieties

Abstract: A smooth projective variety can be viewed as a compact manifold, but its cohomology groups have extra structure: the classes of subvarieties of a given dimension span a convex cone in the relevant (real) cohomology group. For some classes of varieties these cones are easy to compute, but their structure remains mysterious in general. I will explain a conjecture, due to Morrison and Kawamata, that aims to describe these cones for varieties with “nonnegative curvature”. Time permitting I will discuss the case of elliptic Calabi–Yau 3-folds, which may be of interest in theoretical physics.