Moduli Problems in Algebraic Geometry

在这个项目中，研究者研究了代数几何中的两种模空间，一种是稳定态射的模，另一种是稳定层的模。这两个模空间都与超弦理论的研究密切相关。具体而言，研究人员在四个研究方向上工作。第一个是关于虚模圈的有效构造。第二是继续研究模空间的退化，包括稳定层和稳定态射的模。三是研究Calabi-Yau流形上的开弦理论，三是研究稳定层模的一些新性质，这是数学科学的一个分支--代数几何的研究。研究者研究某些空间（数学中称为模空间）的性质，这些空间由可以用代数性质表征的对象组成。(An这些空间很重要，因为它们是高能物理学中超弦理论的组成部分，超弦理论致力于统一量子力学和引力。在过去的十年里，超弦理论的研究提出了许多关于这些空间的挑战性问题，其中一些仍然是开放的。本研究项目将解决其中一些挑战。

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 ofmathematical science. The investigator studies theproperties of certain spaces (called moduli space in mathematics)that consist of objects that can be characterized by algebraicproperties. (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.