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

项目负责人：Ralph L. Cohen, Jun Li, Dennis P. sullivan本项目研究了Riemann曲面的模空间拓扑结构及其在弦拓扑中的应用，以及物理学中弦理论引起的一些数学问题。这是一个涉及代数拓扑、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研究机会：马德森-魏斯对著名的芒福德猜想关于模空间稳定上同调的证明，查斯和沙利文关于流形环空间拓扑结构的发现，以及物理学中开放弦理论的最新进展。该项目的目标之一是理解Madsen和Weiss定理对查斯-沙利文“弦拓扑”理论的影响。本课题的另一个方面是研究代数几何中弦拓扑与Gromov-Wittentheory之间的关系。这个项目的长期目标是研究这个理论如何帮助给出一个数学框架来分析物理学中由开放弦理论引起的某些特定问题。长期以来，几何问题的动机一直是试图理解物理理论。爱因斯坦的广义相对论，以及将其建立在坚实的数学基础上的尝试，在很大程度上推动了整个20世纪微分几何的发展。在本世纪的最后20年里，著名的麦克斯韦电和磁方程的推广带来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。将其置于坚实的数学基础上是相当具有挑战性的，并且激发了相当多的几何新研究。例如，弦理论的技术预测了枚举几何中一些经典问题的答案，其中许多问题后来由数学家M. Gromov和物理学家E. Witten在代数几何中的新理论中得到了验证。弦理论涉及到理解弦如何随着时间的推移而振动。随着字符串的演化，它会绘制出一个二维的“世界表”。因此，弦理论背后的数学必须研究“弦”空间，即曲线和环，以及环境空间中二维表面的空间。这个项目的动机是最近在理解弦空间拓扑结构（“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。该项目的目标是了解这一突破对“弦拓扑”的影响，了解这种拓扑理论与Gromov和Witten的几何理论的关系，并将这些理论应用于物理学中弦理论产生的某些具体问题。该奖项支持斯坦福大学和纽约州立大学石溪分校的一个重点研究小组。

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.