The PI plans to investigate the geometry of various spaces of sheaves on curves and surfaces. In particular, she will study a space of stable sheaf quotients of a trivial rank n sheaf on a curve, as the curve is allowed to vary in the Deligne-Mumford moduli space. The space of stable quotients was constructed earlier in joint work with D. Oprea and R. Pandharipande. In one direction, it may help to clarify features of the tautological ring of the moduli space of curves, to which it admits a proper morphism. In a different direction, the intersection theory of the moduli space of stable quotients has close connections with the Gromov-Witten theory of Grassmannians and complete intersections in Grassmannians, which would be interesting to understand completely. The PI also proposes to continue the study of strange duality phenomena on moduli spaces of sheaves on surfaces. Recently, there has been good progress in this area in the case of rational surfaces and K3 surfaces.
Moduli spaces of sheaves on varieties are geometric objects of great importance in both mathematics and theoretical physics. Mathematically, their study was initiated by Mumford's construction of the moduli space of stable bundles on a smooth projective curve via geometric invariant theory. Later, Donaldson showed that the intersection theory of moduli spaces of rank 2 sheaves on surfaces is a useful tool for understanding the geometry of four-manifolds. In physics, moduli spaces of vector bundles arise naturally in the fundamental investigation of gauge theories. The PI plans to further the study of parameter spaces of sheaves in various settings, and advance our understanding of these important geometric structures.
Moduli spaces of sheaves on varieties are important geometric objects in both mathematics and theoretical physics. Spaces of sheaves over a variety often reflect geometric properties of the underlying variety. In low dimensions, intersection-theoretic properties of spaces of sheaves were carefully studied in the framework of Yang-Mills, Donaldson, and Donaldson-Thomas theories. In physics, moduli spaces of vector bundles arise naturally in the fundamental study of gauge theories. The PI studied parameter spaces of sheaves on curves and surfaces in a variety of settings. Specifically, during the period of support, a geometric symmetry termed "strange duality" was investigated in the case of abelian and K3 surfaces. In joint work with Dragos Oprea, the symmetry was demonstrated for a large class of topological types of the sheaves and generic choices of the surfaces in their own moduli space. Furthermore, the abelian surface case of the duality admits three distinct geometric flavors which were all studied. When the duality is formulated globally over the moduli space of K3 or abelian surfaces, interesting natural vector bundles arise over these classic moduli spaces, whose properties, such as the Chern character, are far from being understood. Progress in this direction would also shed light on the cohomology and the tautological cohomology of these classic moduli spaces. This line of investigation is still quite open.
