Moduli Problems in Algebraic Geometry

本文研究了代数几何中的两类模空间，一类是稳定态射的模，另一类是稳定层的模。这两个模空间都与超弦理论的研究密切相关。具体地说，调查者在四个研究方向上工作。第一个是关于虚拟模圈的有效构造。二是继续研究模空间的退化，包括稳定层的模和稳定态射的模。第三个是研究Calabi-Yau流形上的开弦理论，最后是研究一些关于稳定层的模的新猜想。这个项目是数学科学的一个分支代数几何的研究。研究人员研究某些空间(在数学上称为模空间)的性质，这些空间由可用代数性质刻画的对象组成。这些空间很重要，因为它们是高能物理中超弦理论的一部分，该理论致力于统一量子力学和引力。在过去的十年里，超弦理论的研究对这些空间提出了许多具有挑战性的问题，其中一些仍然是开放的。这项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.