The PI will work on a wide range of research projects related to the geometry and intersection theory of moduli spaces in algebraic geometry.Three main directions are proposed. First, the PI will study Le Potier's strange duality conjecture for spaces of generalized theta functions. Along the way, the author will investigate questions related to the tautological rings of the moduli space of K3 surfaces, abelian surfaces, and Calabi-Yau manifolds, the positivity of the theta linear series, and the Brill-Noether theory of sheaves over higher dimensional bases. A second direction concerns intersection theoretic calculations on the moduli space of stable quotients for local toric and hypersurface geometries, and their comparison with Gromov-Witten theory. Thirdly, the author plans to compute stable pair invariants of local curve and surface geometries by comparison with intersection theory of Quot scheme, a topic which has been at the center of the PI's research in recent years.
The PI works in algebraic geometry, which is the study of solutions of polynomial equations. More specifically, the PI's research focuses on moduli theory and enumerative geometry. Moduli spaces put together objects with similar structure, with the goal of obtaining a global unified view of the structure under study, both qualitatively and quantitatively. This will be beneficial for algebraic geometry, but also for a number of related fields. In recent years, the PI mentored undergraduate and graduate students in areas close to his research, and will continue to do so. He plans to develop new algebraic geometry courses, and to write an undergraduate algebraic geometry text. He will co-organize a seminar at his institution, as well as the Southern California Algebraic Geometry Seminar. Furthermore, the PI will organize workshops focusing on topics related to moduli spaces of sheaves. Finally, the PI will be involved in high school and college competitions, and in the preparation of gifted high school students.
