Award: DMS-0244550 and DMS-0244100 Principal Investigator: Ralph L. Cohen, Jun Li, and Dennis P. Sullivan
This project investigates the topology of moduli spaces of Riemann surfaces, their applications to string topology, and certain mathematical questions arising from string theory in physics. It is a collaborative project involving algebraic topology, algebraic geometry, and Riemann surface theory. It will pursue significant new research opportunities arising from three recent important developments: The Madsen-Weiss proof of the famous conjecture of Mumford on the stable cohomology of moduli spaces, the discovery by Chas and Sullivan of the new structures on the topology of loop spaces of manifolds, and recent advances in open string theory in physics. One of the goals of this project is to understand the implications of Madsen and Weiss' theorem on the Chas-Sullivan "String topology" theory. Another aspect of this project is to study the relationship between string topology and Gromov-Witten theory in algebraic geometry. A longer term goal of this project is to investigate how this theory can help to give a mathematical framework for analyzing certain specific questions motivated by open string theory in physics.
Geometric questions have long been motivated by the attempt to understand physical theories. Einstein's general theory of relativity, and the attempt to place it in firm mathematical foundations, motivated much of the development of differential geometry throughout the 20th century. During the last 20 years of the century generalizations of the famous Maxwell's equations for electricity and magnetism led to new techniques for studying geometry and topology in dimensions three and four. String theory is a relatively new quantum theory of gravity. Placing it in firm mathematical foundations is quite challenging, and has motivated quite a bit of new research in geometry. For example the techniques of string theory predicted the answers to some classical questions in enumerative geometry, many of which were later verified using a new theory in algebraic geometry due to the mathematician M. Gromov, and the physicist, E. Witten. String theory involves understanding how vibrating strings evolve through time. As a string evolves, it maps out a two dimensional "world sheet". So the mathematics behind string theory must study spaces of "strings", or curves and loops, as well as spaces of two dimensional surfaces in an ambient space. This project has been motivated by recent advances in understanding the topological structure of spaces of strings, ("string topology"), as well as a separate breakthrough in understanding the space of two dimensional surfaces. The goal of this project is to understand the implications of this breakthrough on "string topology", understand how this topological theory is related to the geometric theory of Gromov and Witten, and to apply these theories to certain specific questions arising from string theory in physics. This award supports a Focused Research Group based at Stanford University and SUNY at Stony Brook.
