The investigator will study topics in the broad area of enumerative geometry. Quiver varieties form a general type of degeneracy locus, which includes many important varieties such as determinantal varieties and Schubert varieties. The investigator will attempt to generalize known geometric and combinatorial formulas and constructions concerning equioriented quiver varieties of type A to work for more general quiver varieties of Dynkin type. He will also study various cohomology theories of generalized flag manifolds with an emphasis on the multiplication of Schubert classes. In particular, he will attempt to understand the quantum cohomology of isotropic Grassmannians (with A. Kresch and H. Tamvakis). He also hopes to determine if the Gromov-Witten invariants on Grassmannians and other homogeneous spaces are determined by positivity properties. As a separate goal in his research, the investigator will develop computer programs for computing the studied formulas and invariants.
Much development in algebraic geometry has been motivated by enumerative geometric problems, that is, problems that ask for the number of geometric objects of a specified type that satisfy certain conditions. For example, quantum cohomology is a theory for counting the number of curves of given degree in a variety that meet a number of fixed subvarieties. The invention of quantum cohomology was motivated by physics, where the ability to count curves has applications to mirror symmetry. In general, enumerative problems can be attacked by constructing a moduli space that has one point for each geometric object that could be counted, such that the conditions on the objects correspond to subvarieties of the moduli space. This reduces an enumerative problem to understanding the intersection of subvarieties of a larger space. The relevant subvarieties can often be constructed as special cases of quiver varieties, so this type of varieties provide a powerful tool in enumerative geometry. In addition, the study of quiver varieties has given rise to many new relations between algebraic geometry and combinatorics, as well as powerful methods and constructions in both fields.
