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

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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.