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.

本课题是数学科学的一个分支--代数几何的研究。主要研究者（PI）将研究可以由代数性质表征的对象的某些空间（称为模空间）的性质。(An这样的例子是多项式的根）。这些空间描述的解空间是至关重要的研究在许多分支的数学研究和理论物理。PI将致力于几个研究方向。他将致力于全面理解Calabi-Yau三重的高属Gromov-Witten不变量;他还将发展一个关于Calabi-Yau三重的广义Donaldson-Thomas不变量的替代理论;为研究和证明Calabi的BPS态猜想提供了必要的工具，本研究项目是PI长期研究目标的延续，即通过了解新思想来拓宽数学研究，理论物理学，并通过提供对其发展至关重要的数学基础来促进理论物理学的发展。对高亏格GW不变量和DT不变量的研究将从总体上加深对模空间的理解，丰富代数几何的研究。它还将加强代数几何与其他数学学科以及数学物理之间的相互作用。该项目将促进年轻数学研究人员的教学和培训。总的来说，它将为推进该国的科学研究做出贡献。