The PI plans to explore the geometry of parameter spaces of sheaves on algebraic curves and surfaces. She wants to calculate some of the intersection theory, in particular Verlinde numbers, on Gieseker moduli spaces of semistable sheaves on a smooth surface. These intersection-theoretic calculations will help the PI probe whether the level-rank duality which is a feature of moduli spaces of bundles on a curve, can be associated with moduli spaces of sheaves on a surface as well. The author also wishes to explore some of the representation theory of classical and affine Lie algebras on the cohomology of Grothendieck quot schemes on curves, as well as on the cohomology of Gieseker moduli spaces of sheaves on a surface, which would result in a better understanding of the geometric structure of these moduli spaces.
The study of vector bundles in algebraic geometry parallels that of gauge field theories in physics. The most familiar example of a gauge field is the electromagnetic field. Higher-rank vector bundles correspond to nonabelian gauge fields, which appear in high-energy physics, and mathematically are organized as the entries of a matrix. Studying the geometry and intersection theory on the moduli space of bundles, which is the goal of this project, leads in particular to understanding the structure of various integrals on the space of all possible gauge fields. These integrals are mathematical expressions for scattering amplitudes of various physical processes in high-energy physics.
