SM: Geometry and Topology of Moduli Spaces and Applications

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Understanding the geometry and topology of the moduli space of Riemann surfaces and the corresponding mapping class groups has been a goal of central importance in mathematics for many years. In the last 15 years there have been several new perspectives on moduli spaces that have not only increased our understanding of these important objects, but have fundamentally affected major research directions of several areas of topology and geometry, including Hyperbolic Geometry and Geometric Group Theory, Algebraic and Symplectic Geometry, and most recently, Algebraic Topology. In the last five years there have been several startling advances in several of these areas. Taken as a whole, these areas have, in the last few years, represented some of the most exciting directions of study in topology and geometry, and they promise to continue to do so in the forseeable future. This proposal is for the funding of a major, three year emphasis program in the topology and geometry of moduli spaces and related topics. Each year a different mathematical perspective of this topic will be emphasized, but all three years will involve participants from a broad range of subfields. The three areas of emphasis will be Hyperbolic Geometry and Geometric Group Theory, The Algebraic Topology of Moduli Spaces and String Topology, and The Algebraic Geometry of Moduli Spaces and Symplectic Geometry.The study of surfaces has been a major driving force in mathematics since the time of Riemann in the mid 19th century. The space of geometric structures on a given two dimensional surface is known as the moduli space of Riemann surfaces. These moduli spaces have been classically studied in algebraic geometry. With the pioneering work of M. Gromov in the 1970's, these moduli spaces became instrumental in the study of symplectic geometry as well. They are also central in the modern view of low dimensional topology and hyperbolic geometry initiated by Thurston around the same time. With the development of conformal field theory and string theory in the 1980's, these moduli spaces also began to play an important role in theoretical physics. Most recently, techniques of algebraic topology have been brought to bear on the study of these moduli spaces over the last few years with exciting results. Conversely, formalisms from physics and geometry have had a major impact on recent research directions in algebraic topology. The last five years have seen exciting developments in all these geometric and topological areas affecting and affected by moduli spaces of Riemann surfaces. As one can imagine, the excitement produced in these areas of study have attracted many graduate students and young mathematicians. To be effective researchers, it is important that these young mathematicians gain an understanding of the various different perspectives on these moduli spaces and related objects. Cross pollination between these areas both in terms of techniques and directions of research, can have a powerful effect on the development of these central topics in topology and geometry. This proposal is for the funding of a major, three year emphasis program in the topology and geometry of moduli spaces and related topics. The program will be organized by the Mathematics Research Center of Stanford University, one of the leading centers of research in geometry and topology, and by the American Institute of Mathematics, which is a major independent research institute. Each year a different mathematical perspective of this topic will be emphasized, but all three years will involve participants from a broad range of subfields. Some of the world's leading senior mathematicians, their junior colleagues, as well as students will participate in these programs, share and compare their different perspectives and areas of expertise, and will work together to deepen our understanding of this central area of mathematics, and produce new and exciting methods, techniques, and results.