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

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Gromov-维滕K理论的研究中，李教授和他的合作者发现了一些有趣的关系，这些关系涉及数学和物理学的其他领域，包括代数几何、可积系统、表示论和量子几何。 这些调查也自然导致进一步调查格罗莫夫-维滕理论本身。 这个项目的主要重点将是量子上同调，量子K理论和它们的关系，离散KdV层次，户田格，和枚举几何。Gromov-Witten理论是一门新兴的学科，它是数学和理论物理学许多传统分支交叉的产物。 这个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应用数学。 其中一些应用解决了基于传统方法的问题，这些问题被认为是非常困难的。 研究Gromov-维滕理论所用的工具也涉及到许多领域，包括代数几何、拓扑学和分析学。 因此，这是一个正在发展的新领域，它展示了许多丰富的结构，值得进一步研究