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

DMS-0072547袁-Pin Lee量子上同调是光滑射影簇X的普通上同调环的常见杯积的变形。类似地，量子K-理论是X的普通K-环的变形。在他们对Gromov-Witten K-理论的研究中，Lee教授和他的合作者发现了一些与数学和物理中的其他领域的有趣的关系，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。这个项目的主要焦点将集中在量子上同调、量子K理论及其与离散KdV族、Toda格和计数几何的关系上。格罗莫夫-维腾理论是一门新学科，它位于数学和理论物理的许多传统分支的交叉点上。这一理论最初是在弦理论中发现的。此后不久，人们发现了许多数学的新应用。其中一些应用解决了基于传统方法被认为非常困难的问题。用于研究Gromov-Witten理论的工具也涉及许多领域，包括代数几何、拓扑学和分析。因此，这是一个正在发展的新领域，呈现出许多丰富的结构，值得进一步研究。

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