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

摘要奖：DMS-0244550和DMS-0244100主要研究人员：拉尔夫·L·科恩，李军和丹尼斯·P·沙利文这个项目研究黎曼曲面的模空间的拓扑，它们在弦拓扑中的应用，以及物理中弦理论产生的某些数学问题。它是一个涉及代数拓扑学、代数几何和黎曼曲面理论的合作项目。它将从最近的三个重要发展中寻求重要的新的研究机会：著名的Mumford关于模空间的稳定上同调猜想的Madsen-Weiss证明，Chas和Sullivan发现的关于流形的环空间拓扑的新结构，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Madsen和Weiss定理对Chas-Sullivan“弦拓扑”理论的影响。这个项目的另一个方面是研究代数几何中弦拓扑与Gromov-Witten理论之间的关系。这个项目的一个较长期的目标是调查这个理论如何能够帮助给出一个数学框架来分析物理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受到理解物理理论的尝试的激励。爱因斯坦的广义相对论，以及试图将其建立在坚实的数学基础上的尝试，推动了整个20世纪微分几何的发展。在本世纪的最后20年里，著名的麦克斯韦电磁场方程的推广导致了研究三维和四维几何和拓扑的新技术。弦理论是一种相对较新的引力量子理论。把它放在坚实的数学基础上是非常具有挑战性的，并激发了相当多的几何学新研究。例如，弦理论的技术预测了列举几何中一些经典问题的答案，其中许多问题后来被数学家M.Gromov和物理学家E.Witten用代数几何中的一种新理论验证。弦理论涉及理解振动的弦是如何随时间演变的。随着弦的进化，它绘制出一个二维的“世界薄片”。因此，弦理论背后的数学必须研究“弦”的空间，即曲线和环，以及环境空间中二维曲面的空间。这个项目的动机是在理解弦空间的拓扑结构(“弦拓扑”)方面的最新进展，以及在理解二维曲面空间方面的另一项突破。这个项目的目标是理解这一突破对“弦拓扑学”的影响，理解这一拓扑学理论如何与格罗莫夫和维滕的几何理论相联系，并将这些理论应用于物理中弦理论产生的某些具体问题。该奖项支持总部设在斯坦福大学和纽约州立大学石溪分校的一个专注研究小组。

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.