Moduli Problems in Algebraic Geometry, Their Structures and Their Applications

这个项目是对代数几何的研究，它是数学科学的一个分支。首席调查员(PI)将研究对象的某些空间(称为模空间)的性质，这些空间可以用代数性质来表征。(多项式的根就是这样的一个例子)。这些空间描述了对数学研究的许多分支和理论物理中的研究至关重要的解空间。PI将在几个研究方向上开展工作。他将致力于全面理解Calabi-Yau三重高亏格Gromov-Witten不变量；他还将发展一种关于Calabi-Yau三重广义Donaldson-Thomas不变量的替代理论；开发必要的工具来研究和证明关于Calabi-Yau三重BPS态的猜想。本研究项目是Pi长期研究目标的延续，即通过从理论物理中了解新的思想来拓宽数学研究，并通过提供对理论物理的进步至关重要的数学基础来促进理论物理的发展。高亏格GW不变量和DT不变量的研究进展将促进我们对模空间的理解，丰富代数几何的研究。它还将加强代数几何和其他数学学科以及与数学物理之间的互动。该项目将促进教、学和培养年轻的数学研究人员。总体而言，它将为促进国家的科学研究做出自己的贡献。

This project is a research in algebraic geometry, a branch of mathematical science. The Principal Investigator (PI) will study properties of certain spaces (called moduli space) of objects that can be characterized by algebraic properties. (An example of such are the roots of polynomials). These spaces describe solution spaces that are vital to research in many branches of mathematical researches and in theoretical physics. The PI will work on several research directions. He will work toward a full understanding of high genus Gromov-Witten invariants of Calabi-Yau threefolds; he will also develop an alternative theory on generalized Donaldson-Thomas invariants of Calabi-Yau threefold; develop necessarily tools to study and prove the conjecture on BPS-states of Calabi-Yau threefolds.This research project is the continuation of PI's long term research goal of broadening mathematical research by understanding new idea from theoretical physics and contributing to the development of theoretical physics by providing mathematical foundation vital to its advancement. The progress on studying high genus GW invariants and DT invariants will advance our understanding of moduli spaces in general; enrich the research in algebraic geometry. It will also strengthen the interaction between algebraic geometry and other subjects of mathematics, and with mathematical physics. This project will promote teaching, learning and training young mathematical researchers. Over all, it will contribute its share in advancing the science research in the country.