FRG: Collaborative Research: Moduli Spaces of Riemann Surfaces and String Topology

摘要奖：DMS-0244550和DMS-0244100主要研究者：Ralph L. Cohen，Jun Li，and Dennis P. Sullivan本项目研究黎曼曲面的模空间的拓扑结构，它们在弦拓扑中的应用，以及物理学中弦理论引起的某些数学问题。这是一个合作项目，涉及代数拓扑，代数几何和黎曼曲面理论。它将追求重大的新的研究机会，从最近的三个重要发展产生：Mumford关于模空间稳定上同调的著名猜想的Madsen-Weiss证明，Chas和Sullivan关于流形的loop空间拓扑的新结构的发现，以及物理学中开弦理论的最新进展。这个项目的目标之一是理解马德森和韦斯定理对物理学中开弦理论的影响。Chas-Sullivan的“弦拓扑”理论。 本项目的另一个方面是研究弦拓扑与代数几何中Gromov-Witten理论之间的关系。 这个项目的一个长期目标是研究这个理论如何能够帮助给出一个数学框架来分析物理学中由开弦理论激发的某些特定问题。几何问题长期以来一直是由理解物理理论的尝试激发的。 爱因斯坦的广义相对论，以及试图将其置于坚实的物理基础之上的努力，推动了整个世纪微分几何的发展。 在世纪的最后20年里，著名的麦克斯韦电磁方程的推广导致了研究三维和四维几何和拓扑的新技术。 弦理论是相对较新的引力量子理论。 把它建立在坚实的数学基础上是相当具有挑战性的，并激发了相当多的几何学新研究。 例如，弦理论的技巧预言了枚举几何中一些经典问题的答案，其中许多问题都是由数学家M.格罗莫夫和物理学家E.维滕。弦理论涉及到理解振动的弦如何随时间演化。 随着弦的演化，它会映射出一个二维的“世界表”。 所以弦论背后的数学必须研究“弦”的空间，或者曲线和环，以及周围空间中的二维表面的空间。 这个项目的动机是最近在理解弦空间的拓扑结构（“弦拓扑”）方面的进展，以及在理解二维表面空间方面的单独突破。 这个项目的目标是理解这一突破对“弦拓扑”的影响，理解这种拓扑理论与格罗莫夫和维滕的几何理论的关系，并将这些理论应用于物理学中弦理论所产生的某些特定问题。 该奖项支持斯坦福大学和纽约州立大学斯托尼布鲁克分校的重点研究小组。

AbstractAward: DMS-0244550 and DMS-0244100Principal Investigator: Ralph L. Cohen, Jun Li, and Dennis P. SullivanThis project investigates the topology ofmoduli spaces of Riemann surfaces, their applications to stringtopology, and certain mathematical questions arising from stringtheory in physics. It is a collaborative project involvingalgebraic topology, algebraic geometry, and Riemann surfacetheory. It will pursue significant new research opportunitiesarising from three recent important developments: TheMadsen-Weiss proof of the famous conjecture of Mumford on thestable cohomology of moduli spaces, the discovery by Chas andSullivan of the new structures on the topology of loop spaces ofmanifolds, and recent advances in open string theory in physics.One of the goals of this project is to understand theimplications of Madsen and Weiss' theorem on the Chas-Sullivan"String topology" theory. Another aspect of this project is tostudy the relationship between string topology and Gromov-Wittentheory in algebraic geometry. A longer term goal of this projectis to investigate how this theory can help to give a mathematicalframework for analyzing certain specific questions motivated byopen string theory in physics.Geometric questions have long been motivated by the attempt tounderstand physical theories. Einstein's general theory ofrelativity, and the attempt to place it in firm mathematicalfoundations, motivated much of the development of differentialgeometry throughout the 20th century. During the last 20 yearsof the century generalizations of the famous Maxwell's equationsfor electricity and magnetism led to new techniques for studyinggeometry and topology in dimensions three and four. Stringtheory is a relatively new quantum theory of gravity. Placing itin firm mathematical foundations is quite challenging, and hasmotivated quite a bit of new research in geometry. For examplethe techniques of string theory predicted the answers to someclassical questions in enumerative geometry, many of which werelater verified using a new theory in algebraic geometry due tothe mathematician M. Gromov, and the physicist, E. Witten.String theory involves understanding how vibrating strings evolvethrough time. As a string evolves, it maps out a two dimensional"world sheet". So the mathematics behind string theory muststudy spaces of "strings", or curves and loops, as well as spacesof two dimensional surfaces in an ambient space. This projecthas been motivated by recent advances in understanding thetopological structure of spaces of strings, ("string topology"),as well as a separate breakthrough in understanding the space oftwo dimensional surfaces. The goal of this project is tounderstand the implications of this breakthrough on "stringtopology", understand how this topological theory is related tothe geometric theory of Gromov and Witten, and to apply thesetheories to certain specific questions arising from string theoryin physics. This award supports a Focused Research Group basedat Stanford University and SUNY at Stony Brook.