Moduli of surfaces, vector bundles, and mirror symmetry

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The PI has established a connection between the classification of stable vector bundles on an algebraic surface and the compactification of the moduli space of deformations of the surface. The PI will study this correspondence in detail for surfaces of general type, in particular, its relation to the Donaldson theory of instanton invariants of smooth 4-manifolds. Jointly with Mark Gross and Sean Keel, the PI has described an explicit construction of the mirror partner to a non-compact Calabi--Yau manifold of complex dimension 2. The PI will pursue two related projects: an algebraic description of the symplectic cohomology ring of a non-compact Calabi-Yau manifold, and the construction of a canonical basis of global sections of an ample line bundle on a K3 surface, analogous to theta functions for polarized abelian varieties.The PI will study the classification of certain 4-dimensional geometric spaces and the ways in which such a space can be continuously deformed or undergo a "degeneration" given by a subset of the space collapsing to a point. In prior work the PI related such degenerations to the classification of bundles of linear spaces over the given geometric space. He will study this correspondence in detail, in particular its connection with work of Donaldson motivated by theoretical physics. Mirror symmetry is a mysterious correspondence between pairs of geometric spaces called Calabi-Yau manifolds arising in string theory. The PI will pursue two projects inspired by mirror symmetry. The first is a description in explicit terms of an algebraic structure built from counts of area-minimizing surfaces inside a Calabi-Yau manifold. The second is the construction of natural functions on Calabi-Yau manifolds. In the simplest case of a torus (the surface of a donut) these functions were known classically and are important in many areas of mathematics.