Moduli Problems in Algebraic Geometry, Their Structures and Their Applications

代数几何中的模问题、其结构及其应用

• 批准号: 1104553
• 负责人: Jun Li
• 金额: $ 48万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1104553&HistoricalAwards=false
• 关键词: Moduli; Problems; Algebraic; Geometry; Their

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.

该项目是数学科学的一个代数几何形状的研究。主要研究者（PI）将研究可以用代数特性来表征的某些空间（称为模量空间）的特性。 （这样的例子是多项式的根）。这些空间描述了在数学研究和理论物理学的许多分支中研究至关重要的解决方案空间。 PI将在多个研究方向上工作。他将致力于对Calabi-yau三倍的高级格罗莫夫（Gromov-witten）属不变属的高级理解；他还将开发出关于Calabi-yau三倍的广义Donaldson-Thomas不变的替代理论。开发必要的工具来研究和证明Calabi-yau三倍的BPS状态的猜想。这项研究项目是PI的长期研究目标的延续，即从理论物理学中理解新思想，并通过为其进步提供数学基础，从而扩大数学研究的长期研究目标。研究高属GW不变性和DT不变性的进展将提高我们对Moduli空间的理解。丰富了代数几何形状的研究。它还将加强代数几何形状与数学的其他主题以及数学物理学之间的相互作用。该项目将促进教学，学习和培训年轻的数学研究人员。总体而言，它将在推进该国的科学研究方面做出贡献。