Moduli Problems in Algebraic Geometry

Stanford University, Stanford CA

Investigators

Abstract

In this project, the investigator investigates two kinds of moduli spaces in algebraic geometry, one is the moduli of stable morphisms and the other is the moduli of stable sheaves. Both moduli spaces are closely related to the research in Super- String theories. Specifically, the investigator works on four research directions. The first is on the effective construction of virtual moduli cycles. The second is to continue working on degeneration of the moduli spaces, including the moduli of stable sheaves and stable morphisms. The third is to study open string theory in Calabi-Yau manifolds and the last to study some new conjectures on the moduli of stable sheaves. This project is a research in algebraic geometry, a branch of mathematical science. The investigator studies the properties of certain spaces (called moduli space in mathematics) that consist of objects that can be characterized by algebraic properties. (An example is the roots of polynomials). These spaces are important because they are part of the building blocks of the Super-String theory in high energy physics, a theory devoted to unify quantum mechanic and gravity. In the last decade, research in Super-String theory raised many challenging questions about these spaces, some remains open. This research project will address some of these challenge.