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

关于穿孔表面的修饰的Teichmueller理论给经典的Teichmueller理论提供了一个新的观点，而且，这一工作与高能物理的弦理论之间有联系。在此基础上还发展出了关于圆的保向同胚群（即我们的普适Teichmueller空间模型）中Moebius群右上集空间的丰富几何理论，以及一个与（Richard） Thompson群同构的普适映射类群。在代数数论中，这个普遍理论也有不同的、有趣的应用，用于各种看似不同的主题（例如，马尔可夫元组、二次形式的高斯积和格罗腾迪克版本的绝对伽罗瓦理论）。目前的研究旨在进一步发展这些观点，并对这些联系进行更深入的研究。在一个固定的二维曲面S上，关于所有可能的几何结构的空间T(S)，在数学中有一些重要而可敬的问题。想象一下，取一个曲面（比如气球的表面），在上面刺一个洞；这个洞，即缺失点，被称为穿孔，在过去十年左右的时间里，有一些特殊的技术用于研究穿孔表面S上几何结构的空间T(S)。值得注意的是，粒子物理弦理论中的各种量是由穿孔表面S的这些空间T(S)的显式几何不变量给出的。此外，这些技术导致了对数论中各种经典数学结构的新见解。本课题的研究重点在于这些代数应用。＊＊＊

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. ***