Formulas for Quiver Varieties and Quantum Schubert Calculus

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The investigator will study topics in the broad area of enumerativegeometry. Quiver varieties form a general type of degeneracy locus,which includes many important varieties such as determinantalvarieties and Schubert varieties. The investigator will attempt togeneralize known geometric and combinatorial formulas andconstructions concerning equioriented quiver varieties of type A towork for more general quiver varieties of Dynkin type. He will alsostudy various cohomology theories of generalized flag manifolds withan emphasis on the multiplication of Schubert classes. In particular,he will attempt to understand the quantum cohomology of isotropicGrassmannians (with A. Kresch and H. Tamvakis). He also hopes todetermine if the Gromov-Witten invariants on Grassmannians and otherhomogeneous spaces are determined by positivity properties. As aseparate goal in his research, the investigator will develop computerprograms for computing the studied formulas and invariants.Much development in algebraic geometry has been motivated byenumerative geometric problems, that is, problems that ask for thenumber of geometric objects of a specified type that satisfy certainconditions. For example, quantum cohomology is a theory for countingthe number of curves of given degree in a variety that meet a numberof fixed subvarieties. The invention of quantum cohomology wasmotivated by physics, where the ability to count curves hasapplications to mirror symmetry. In general, enumerative problems canbe attacked by constructing a moduli space that has one point for eachgeometric object that could be counted, such that the conditions onthe objects correspond to subvarieties of the moduli space. Thisreduces an enumerative problem to understanding the intersection ofsubvarieties of a larger space. The relevant subvarieties can oftenbe constructed as special cases of quiver varieties, so this type ofvarieties provide a powerful tool in enumerative geometry. Inaddition, the study of quiver varieties has given rise to many newrelations between algebraic geometry and combinatorics, as well aspowerful methods and constructions in both fields.