Mathematical Sciences: Geometry and Arithmetic of Riemann's Moduli Space

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9610041 Penner The decorated Teichmueller theory of punctured surfaces gives a new point of view on the classical Teichmueller theory, and there are, moreover, connections between this work and the string theory of high-energy physics. There has also evolved from this a rich geometric theory of the space of right cosets of the Moebius group in the group of orientation-preserving homeomorphisms of the circle (which is our model of a universal Teichmueller space) as well as a sensible universal mapping class group, which turns out to be isomorphic to the (Richard) Thompson group. There are also different and interesting applications of this universal theory to various seemingly disparate topics in algebraic number theory (such as Markov tuples, the Gauss product of quadratic forms, and Grothendieck's version of absolute Galois theory, for instance). Current research aims to develop these ideas further and pursue a deeper study of these connections. There are important and venerable questions in mathematics about the space T(S) of all possible geometric structures on a fixed two-dimensional surface S. Imagine taking a surface (like the surface of a balloon, for instance) and pricking a hole in it; the hole, i.e., the missing point, is called a puncture, and there are special techniques developed over the last decade or so for studying the space T(S) of geometric structures on a punctured surface S. Remarkably enough, various quantities in the string theory of particle physics are given by explicit geometric invariants of these spaces T(S) for punctured surfaces S. Moreover, these techniques lead to new insights about various classical mathematical constructions in number theory, and this project's research emphasis lies in these algebraic applications. ***