The PI of this project will study questions related to the moduli spaces of sheaves and morphisms. Regarding the moduli spaces of sheaves, the main themes to be covered are: the 'strange duality' isomorphism for the moduli space of bundles on a curve as the curve varies in moduli - in particular, extensions of 'strange duality' to nodal curves and to general structure groups; the Theta bundles, their sections, and 'rank-level' dualities on the moduli spaces of sheaves on surfaces (the latter in collaboration with Alina Marian); computation of intersection theoretic invariants on the moduli space of higher-rank framed/unframed sheaves on threefolds. Regarding stable morphisms, and building on previous work, the author will pursue a further study of the topology of the moduli space of stable maps and of the associated tautological systems. The projects above are aimed to the understanding of the topology, geometry and intersection theory of various moduli spaces of sheaves and morphisms. One of the goals is to exploit the inter-connections these moduli spaces enjoy.
Moduli spaces arise as answers to classification and parametrization problems, putting together objects with similar structure. Questions about these objects (e.g. counting problems, questions about families) often involve a detailed study of the relevant moduli spaces. The moduli spaces of stable sheaves and stable morphisms are two of the primary examples occurring in algebraic geometry. In addition to their importance within the field, they bear multiple connections with other branches of classical and modern mathematics such as complex analysis, representation theory, theoretical physics, symplectic geometry, gauge theory, differential geometry, combinatorics and topology.
