Colloquium

Gromov-Witten invariants and lattice paths in polygons

by Grigory Mikhalkin, Associate Professor of Mathematics, University of Utah, Salt Lake City

Abstract

The talk presents a new formula for the Gromov-Witten invariantsof arbitrary genus in the projective plane, the hyperboloid andother similar surfaces. From a mathematical point of view theseinvariants count the number of holomorphic curves with givennumerical characteristics (such as the genus and the degree)passing through a collection of generically chosen points.

It turns out that such invariants can be computed by means of certainlattice paths connecting two vertices of a certain polygon. Eachrelevant path will correspond to one or more holomorphic curves inthe so-called "large complex limit".