Quantum Cohomology, Quantum K-theory, Frobenius Manifolds and Integrable Systems

项目负责人：李元平，主要研究Gromov—Witten理论及其在数列几何、可积系统、k理论、表示理论等数学领域的关系和应用。重点放在这些领域相交的一些关键问题上。这些问题包括高自旋曲线的研究、量子k理论、计数测量学和Frobenius流形。格罗莫夫—维滕理论诞生于20世纪90年代初，是数学与物理相互作用的产物。在数学方面，它始于Gromov发明的一种产生辛不变量的新方法。在物理方面，是威滕对弦理论的研究。虽然Gromov- Witten理论的历史相对较短，但它是一个重要的活跃研究领域。这不仅是因为它对老问题的强大解决方案，而且还因为它的跨学科性质，与数学和弦理论的几个领域的持续投入。本项目拟从多个角度研究Gromov—Witten理论，并探索其在其他学科领域的应用。

AbstractAward: DMS-0305895Principal Investigator: Yuan-Pin LeeThe main focus of this project is to study Gromov--Witten theoryand its relations and applications to other areas of mathematics,including enumerative geometry, integrable systems, K-theory, andrepresentation theory. The emphasis is placed on some keyproblems where these areas intersect. These problems include thestudy of higher spin curves, quantum K-theory, enumerativegeometry, and Frobenius manifolds.Gromov--Witten theory was born in early 1990's through theinteraction of mathematics and physics. On the mathematical side,it starts with Gromov's invention of a new way to producesymplectic invariants. On the physical side, it was Witten'sstudy of string theory. Although of a relatively short history,Gromov--Witten theory is a major active field of research. Thisis due not only to its powerful solutions to old problems, butalso to its interdisciplinary nature, with continuing inputs fromseveral fields of mathematics and string theory. This projectproposes to investigate Gromov--Witten theory through variousperspectives and to search for its applications in other subjectareas.