Three projects in the theory of moduli in algebraic geometry are addressed in this proposal. The recent considerable influence of string theory on algebraic geometry has led to the introduction of new moduli spaces to study, new techniques for studying them, and interesting ``quantum'' invariants defined in terms of them. This proposal is inspired by the physics, but remains firmly grounded in algebraic geometry.
Moduli spaces are spaces that parametrize geometric objects. The first project deals with the spaces parametrizing holomorphic maps from the Riemann sphere to a complex projective manifold, and the ``Gromov-Witten'' invariants that can be computed as integrals on such spaces. The second deals with the spaces parametrizing holomorphic correspondences (i.e. Riemann surfaces lying over the Riemann sphere together with a holomorphic map to a complex projective manifold) as an alternative source of Gromov-Witten invariants for Riemann surfaces of positive genus. The last project is an investigation of moduli spaces generalizing the spaces of holomorphic vector bundles on an algebraic surface.
