Moduli Problems in Algebraic Geometry

本课题研究了代数几何中的两类模空间，一类是稳定态射的模空间，另一类是稳定束的模空间。这两个模空间都与超弦理论的研究密切相关。具体而言，研究者在四个研究方向上工作。第一部分是关于虚模环的有效构造。二是继续研究模空间的退化，包括稳定束和稳定态射的模。第三部分研究了Calabi-Yau流形中的开放弦理论，最后研究了稳定轴的模的一些新猜想。这个项目是对代数几何的研究，代数几何是数学科学的一个分支。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。（一个例子是多项式的根）。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。在过去的十年中，超弦理论的研究提出了关于这些空间的许多具有挑战性的问题，其中一些仍然是开放的。本研究项目将解决其中的一些挑战。

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.