Algebraic Geometry Inspired by Physics

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