Gromov-Whitten Theory and its Relations With K-Theory, Integrable Systems and Enumerative Geometry

Gromov-Whitten 理论及其与 K 理论、可积系统和枚举几何的关系

• 批准号: 0072547
• 负责人: Yuan-Pin Lee
• 金额: $ 8.13万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072547&HistoricalAwards=false
• 关键词: Gromov; Whitten; Theory; its; Relations

DMS-0072547Yuan-Pin LeeQuantum cohomology is a deformation of usual cup productof ordinary cohomology ring of a smooth projective variety X. Similarly quantum K-theory is a deformation of ordinary K-ring of X. In their study of Gromov--Witten K-theory, Professor Lee and his collaborators have found some interesting relations to other fields in mathematics and physics including algebraic geometry, integrable systems, representation theory and quantum geometry. These investigations also naturally lead to further investigations of Gromov--Witten theory itself. The main focus of this project will be on quantum cohomology, quantum K-theory and their relations to discrete KdV hierarchies, Toda lattices, and enumerative geometry. Gromov-Witten theory is a new subject that lies within theintersection of many traditional branches of mathematics and theoretical physics. This theory was originally discovered within string theory. Shortly thereafter people found many new applications to mathematics. Some of these applications solvedproblems that, based on traditional methods, were considered tobe very difficult. The tools used to study Gromov--Witten theory also involves many fields, including algebraic geometry, topology and analysis. Thus, this is a growing new field which exhibits many rich structures and deserves further investigation

DMS-0072547Yuan-Pin Lee量子上同调是光滑射影簇 X 的普通上同调环的普通杯积的​​变形。类似地，量子 K 理论是 X 的普通 K 环的变形。在他们的 Gromov--Witten K- 的研究中在理论方面，李教授和他的合作者发现了与数学和物理学其他领域的一些有趣的关系，包括代数几何、可积系统、表示论和量子几何。 这些研究自然也导致了对格罗莫夫-维滕理论本身的进一步研究。 该项目的主要重点是量子上同调、量子 K 理论及其与离散 KdV 层次结构、Toda 晶格和枚举几何的关系。格罗莫夫-维滕理论是一门新学科，位于数学和理论物理学许多传统分支的交叉点。 该理论最初是在弦理论中发现的。 此后不久，人们发现了许多新的数学应用。 其中一些应用程序解决了基于传统方法被认为非常困难的问题。 用于研究格罗莫夫-维滕理论的工具还涉及许多领域，包括代数几何、拓扑和分析。 因此，这是一个不断发展的新领域，具有许多丰富的结构，值得进一步研究