Robert Penner, Ph.D. - US grants

Robert Penner will study decorated Teichmuller theory in several directions, as follows. Integrals of monomials of Miller- Morita-Mumford classes over the moduli space of Riemann surfaces should be computable using intersection theory on an appropriate V- manifold compactification of moduli space; in particular, all the necessary ingredients seem to be in place for applying this scheme to the computation of Weil-Petersson volumes for once-punctured surfaces. There is also a "universal" analog of the decorated Teichmuller space, and essentially all of the classical structures extend to this setting; he hopes the study of this universal space will shed light on the classical theory. He furthermore hopes to compute the complex structure of moduli in his coordinates. Computer work will be pursued to derive the combinatorial structure of the moduli space of once-punctured surfaces of genus three, and he hopes explicitly to compute the homology of this spaces. Finally, a program has evolved for extending some of his work to the setting of closed surfaces. Teichmuller theory is a seventy year-old blend of algebraic geometry, differential geometry, and topology. As such, advances will interest a broad spectrum of mathematicians. Some also have a bearing on string theory, so some theoretical physicists have a stake in this research as well.

The Department of Mathematics at the University of Southern California will purchase multiprocessing computation servers which will be dedicated to support research in the mathematical sciences. The equipment will be used for several research projects, including in particular: simulating biological neural networks, computing homological invariants of moduli space, interacting particle systems as models of catalytic surface reactions, nonparametric estimation of probability distributions from incomplete data.

9322042 Penner The investigator will continue his study of the geometry and topology of the moduli space of Riemann surfaces. Specifically, he will study a natural simplicial completion of Teichmueller space, where the associated compactification of moduli space is conjectured to be an orbifold; this compactification should be a useful tool for several enumerative problems in geometry. Various arithmetic aspects of moduli space will be studied: polylogarithm identities should be derived from canonical forms on moduli space; there should be a map from moduli space to Volodin space relating the cohomology of moduli space with algebraic K-theory; number fields associated to fatgraphs a la Grothendieck's dessin d'enfant should be calculated for the investigator's existing database of fatgraphs. This same database should be used to calculate homology groups of various moduli spaces on the computer. There are, moreover, many outstanding analytic and arithmetic questions to be studied about his universal Teichmueller space, which is simply the space of orientation-preserving homeomorphisms of the circle modulo the Moebius group. Several of the projects above are closely related to various developments in physics. A basic object in mathematics, called the "moduli space of Riemann surfaces," has been studied for centuries. Roughly, this moduli space is the collection of all possible geometric shapes on a fixed two-dimensional surface. In the last decade or so, our understanding of moduli space has increased, both because of developments in mathematics, as well as new and exciting interfaces with high-energy physics. Indeed, many idealized but interesting real physics problems amount to questions about the geometry of moduli space itself, and this has provided a remarkable influx of techniques, questions, and ideas, both to mathematics and to physics. In this project, the investigator will continue his ongoing mathematical study of moduli space and furt her study certain of these interfaces with high-energy physics. ***

ABSTRACT Malikov 97-01589 The three topics of the proposal are: Harish-Chandra bimodules over affine Lie algebras, localization of admissible representations to the moduli of vector bundles and a Lie algebra of the group of homeomorphisms of the circle. The main tools to be employed are the Bernstein-Beilinson technique of localization of g-modules and related ideas originating in modern physics, for example Kazhdan-Lusztig tensoring and vertex operator algebras The last decade has seen a spectacular interplay of ideas coming from quantum physics and mathematics. This proposal is about the fusion of these ideas. The results might find application in conformal field theory, representation theory, geometry and low dimensional topology.