Genus-zero Gromov-Witten invariants of a smooth variety or orbifold X are rational numbers that -- roughly -- count rational curves on X satisfying certain incidence conditions. They are defined as intersection numbers on a certain moduli space, and many individual Gromov-Witten invariants can be computed explicitly. Mirror symmetry is a beautiful conjecture about the structure of all genus-zero Gromov-Witten invariants of X, which says that they are determined by a very explicit formula. The conjecture has been proved for toric varieties and toric stacks (and certain complete intersections in them). The proofs depend on the fact that these spaces have torus actions with isolated fixed points and 1-dimensional orbits. We prove the conjecture in the case X=Sym^d(P^r), whose 1-dimensional torus orbits are not isolated, by evaluating certain integrals (trivial in the toric case) over Losev-Manin spaces of rational curves.
Speaker(s): Rob Silversmith (Michigan)