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.

该项目是对代数几何（数学科学的一个分支）的研究。首席研究员 (PI) 将研究可以用代数属性来表征的对象的某些空间（称为模空间）的属性。 （这样的一个例子是多项式的根）。这些空间描述了对于数学研究和理论物理的许多分支的研究至关重要的解空间。 PI 将致力于多个研究方向。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卡拉比-丘三重 BPS 态猜想。该研究项目是 PI 长期研究目标的延续，即通过理解理论物理的新思想来拓宽数学研究并为理论物理的发展做出贡献通过提供对其进步至关重要的数学基础。研究高属 GW 不变量和 DT 不变量的进展将增进我们对模空间的一般理解；丰富了代数几何的研究。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、学习和培训。总的来说，它将为推动国家科学研究做出自己的贡献。