PECASE: Intersection Theory On Moduli Spaces

Stanford University, Stanford CA

Investigators

Abstract

Proposal Title: PECASE: Intersection Theory On Moduli Spaces Institution: Stanford University Proposal ID: 0238532 Complicated geometric objects often have a great deal of subtle structure. ``Moduli spaces'' for these objects in some sense capture this structure in a nice package. Properties of moduli spaces are ``universal facts'' about the objects in question. Ideas behind moduli spaces are quite old, dating back to the nineteenth century (at least). In the last thirty years, we have learned a powerful way of studying moduli spaces, thanks to the insights of Grothendieck's school. The last decade has opened up powerful new ways of understanding these spaces. Surprisingly, the impetus often came from other fields, such as theoretical physics or combinatorics. This proposal seeks to approach many pressing problems using techniques from algebraic geometry, highly motivated by insights from other fields. The results in turn should have strong applications in other fields. The investigator also seeks to attract talented high school and undergraduate students into the mathematical sciences, by exposing them to exciting and advanced yet accessible ideas, for example through problem solving; this will be done primarily through the Stanford University Math Camp, a problem solving seminar at Stanford, the Berkeley Math Circle, and various writings. In particular, the goal is to attract students from previously untapped pools of talent. Second, at the graduate level, the investigator will build a center for algebraic geometry at Stanford, by providing resources for graduate students and postdoctoral students, developing new courses, inviting visitors, and sponsoring seminars and conferences, often jointly with other institutions. Third, the investigator will continue to bring sophisticated mathematical ideas (of all levels) to a wider audience through expository writing. The investigator is an algebraic geometer whose primary interest is in intersection theory on moduli spaces. The investigator's goal is to approach many open and classical questions in geometry and related fields using both insights from other fields and modern machinery. The investigator proposes to broaden and deepen his research, by undertaking two longer-term projects, dealing with two of the most important moduli spaces in mathematics: the moduli space of curves, and the Grassmannian and its generalizations. The first project will use modern techniques to illuminate the conjectural and known combinatorial structure behind the ``geometrically natural'' part of the cohomology (or Chow) ring of moduli space of curves (the ``tautological ring''). The second project will use algebro-geometric ideas to solve classical open questions about the structure (algebraic, arithmetic, geometric, enumerative, and more) behind Littlewood-Richardson rules, Schubert problems, and generalizations to other groups. The first project relates to physics, topology, combinatorics, integrable systems, and symplectic geometry; the second involves combinatorics, representation theory, and arithmetic geometry. Thus both will involve developing a base of knowledge in a broad array of different fields, as well as lifelong working relationships and collaborations with researchers in these fields. This project was originally funded as a CAREER award, and was converted to a Presidential Early Career Award for Engineers and Scientists (PECASE) award in September 2004.