Kiumars Kaveh, University of Pittsburgh

Title: Convex bodies in algebraic geometry and symplectic geometry

Abstract:

We start by discussing some basic facts about asymptotic behavior of semigroups of lattice points (which is combinatorial in nature). We will see how this allows one to assign convex bodies to projective algebraic varieties encoding important "intersection theoretic" data. Applying inequalities from convex geometry to these bodies (e.g. Brunn-Minkowski) one immediately obtains Hodge inequalities from algebraic geometry. This is in the heart of theory of Newton-Okounkov bodies. It generalizes the extremely fruitful correspondence between toric varieties and convex polytopes, to arbitrary varieties. We then discuss connection with symplectic (and Kahler) geometry and in particular regarding these bodies as images of moment maps for Hamiltonian torus actions. For "spherical varieties" (or "multiplicity-free spaces") these constructions become very concrete and they bring together algebraic geometry, symplectic geometry and representation theory. For the most part the talk is accessible to anybody with just a basic knowledge of algebra and geometry