Moduli Spaces in Algebraic Geometry, their Structures and their Applications

Stanford University, Stanford CA

Investigators

Abstract

The goal of this project is to investigate the geometry of moduli spaces in algebraic geometry. These spaces are important because they often play fundamental roles in the research of many branches of mathematics and in Super-String theories. Specifically, the Principal Investigator will work on moduli of stable maps and of stable sheaves on algebraic varieties. For the first, he will continue his earlier work on developing an effective theory of high genus Gromov-Witten invariants of Calabi-Yau threefolds, a research direction directly influenced by Super String theory. For the moduli of sheaves, he will continue his work on investigating sheaves on surfaces, aiming to uncover new affine Lie algebra representations based on moduli of sheaves; he will develop a theory on degeneration of moduli of stable sheaves and the associated degeneration formula of their associated Donaldson type invariants. The later will open a new channel to the research of moduli of stable sheaves. In a research field bordering both, he will follow the recent leads from Super-String theory to investigate the BPS-state conjecture of Gopakumar-Vafa; this research in the long run will reveal a direct relation between moduli of sheaves with moduli of stable maps, and thus with that of curves, from a completely new horizon. In the long run, this project will serve to broaden mathematical research by understanding new ideas from theoretical physics and to contribute to the advancement of Super String theories by providing necessary mathematical foundation for it. The listed individual projects are part of research in algebraic geometry, an active research branch in mathematics whose goal is to investigate the geometry, topology and arithmetic property of varieties that are solutions to polynomials. Since last decade, one research field in this branch progressed tremendously in part due to its application to theoretical physics and other fields in general science-it is to investigate moduli spaces of objects studied in algebraic geometry. Of many, a simple minded example is the space of all possible shapes of the surfaces of donuts. Another comparison is like studying space of all possible trajectories of a particle in space. This research project will work on two of the most important moduli spaces in algebraic geometry-moduli of maps and moduli of sheaves; it will also work to explore the connection between these two moduli spaces, eventually developing a connection that will tie these spaces together. Some problems to be investigated are directly from research in theoretical physics, thus the potential impact of this project will go beyond mathematical research. This project will also promote teaching, learning and training students in many ways.