Moduli Spaces in Algebraic Geometry, their Structures and their Applications

代数几何中的模空间、结构及其应用

• 批准号: 0601002
• 负责人: Jun Li
• 金额: $ 50.36万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0601002&HistoricalAwards=false
• 关键词: Moduli; Spaces; Algebraic; Geometry; their

The goal of this project is to investigate the geometry of moduli spaces in algebraic geometry. These spaces are important because they often play fundamental roles in the research of many branches of mathematics and in Super-String theories. Specifically, the Principal Investigator will work on moduli of stable maps and of stable sheaves on algebraic varieties. For the first, he will continue his earlier work on developing an effective theory of high genus Gromov-Witten invariants of Calabi-Yau threefolds, a research direction directly influenced by Super String theory. For the moduli of sheaves, he will continue his work on investigating sheaves on surfaces, aiming to uncover new affine Lie algebra representations based on moduli of sheaves; he will develop a theory on degeneration of moduli of stable sheaves and the associated degeneration formula of their associated Donaldson type invariants. The later will open a new channel to the research of moduli of stable sheaves. In a research field bordering both, he will follow the recent leads from Super-String theory to investigate the BPS-state conjecture of Gopakumar-Vafa; this research in the long run will reveal a direct relation between moduli of sheaves with moduli of stable maps, and thus with that of curves, from a completely new horizon. In the long run, this project will serve to broaden mathematical research by understanding new ideas from theoretical physics and to contribute to the advancement of Super String theories by providing necessary mathematical foundation for it.The listed individual projects are part of research in algebraic geometry, an active research branch in mathematics whose goal is to investigate the geometry, topology and arithmetic property of varieties that are solutions to polynomials. Since last decade, one research field in this branch progressed tremendously in part due to its application to theoretical physics and other fields in general science-it is to investigate moduli spaces of objects studied in algebraic geometry. Of many, a simple minded example is the space of all possible shapes of the surfaces of donuts. Another comparison is like studying space of all possible trajectories of a particle in space. This research project will work on two of the most important moduli spaces in algebraic geometry-moduli of maps and moduli of sheaves; it will also work to explore the connection between these two moduli spaces, eventually developing a connection that will tie these spaces together. Some problems to be investigated are directly from research in theoretical physics, thus the potential impact of this project will go beyond mathematical research. This project will also promote teaching, learning and training students in many ways.

该项目的目的是研究代数几何形状中模量空间的几何形状。这些空间很重要，因为它们经常在数学和超弦理论的许多分支研究中扮演基本角色。具体而言，首席研究人员将在代数品种上处理稳定地图和稳定滑轮的模量。首先，他将继续他的较早的工作，以发展有效的Gromov-witten of Calabi-yau三倍的高级理论，这是一个直接受超弦理论影响的研究方向。对于系束束带的模量，他将继续研究在表面上调查带束的束带，旨在揭示基于滑轮模量的新仿射代数代数。他将发展一个关于稳定滑轮模量变性的理论，以及其相关唐纳森类型不变的相关变性公式。后来的将为稳定滑轮模量研究的新渠道打开新的渠道。在两者接壤的研究领域，他将遵循超级弦理论的最新潜在客户，以研究Gopakumar-Vafa的BPS状态猜想。从长远来看，这项研究将揭示带有稳定地图模束的带模束模量的直接关系，从而从全新的地平线上进行了曲线。从长远来看，该项目将通过理解理论物理学的新思想来扩大数学研究，并通过为其提供必要的数学基础来为超级弦理论的发展做出贡献。列出的单个项目是代数几何的一部分，是代数的研究的一部分，该研究是一个积极的研究分支，在数学领域的积极研究分支，其目的是研究质量，拓扑，拓扑，多种多样的属性，而不是多种多样。自上十年以来，该分支机构的一个研究领域的发展幅度很大，部分原因是它将其应用于理论物理学和一般科学领域的其他领域，这是为了研究代数几何形状研究的对象的模量空间。在许多人中，一个简单的例子是甜甜圈表面所有可能形状的空间。另一个比较就像研究空间中粒子的所有可能轨迹的空间一样。该研究项目将在带束的地图和模量的代数几何形状模型中使用两个最重要的模量空间。它还将有效探索这两个模量空间之间的连接，最终开发出将这些空间绑在一起的连接。需要研究的一些问题直接来自理论物理学的研究，因此该项目的潜在影响将超越数学研究。该项目还将以多种方式促进教学，学习和培训学生。