Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Asymptotic invariants of line bundles and convex bodies by Robert LAZARSFELD, University of Michigan Abstract: It is a classical that hyperplane sections of projective varieties satisfy many beautiful geometric, numerical and cohomological properties. On the other hand, examples due to Cutkosky and others led to the traditional belief that the behavior of more general divisors was mired in pathology. However it has recently become clear that arbitrary effective (or ''big'') divisors display a surprising number of properties analogous to those of ample line bundles. The key is to study the divisors in question from an asymptotic perspective. I'll give an introduction to this circle of ideas, focusing on one invariant (the ''volume'') that measures the rate of growth of the number of sections of powers of a line bundle. I'll also discuss a construction, essentially due to Okounkov, that realizes these invariants as volumes of convex bodies.