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

9610041 Penner穿透表面的装饰Teichmueller理论对经典Teichmueller理论提出了一个新的观点，而且这项工作与高能物理的弦理论之间也有联系。由此，在圆的保定向同胚群(这是我们的普适Teichmueller空间的模型)中的Moebius群的右陪集空间的丰富的几何理论以及一个合理的普适映射类群，证明了它与(Richard)Thompson群同构。这一普适理论在代数数论中的各种看似不同的主题中也有不同和有趣的应用(例如，马尔可夫元组，二次型的高斯积，以及格罗森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想法，并对这些联系进行更深入的研究。关于固定的二维曲面S上所有可能的几何结构的空间T(S)，在数学中有一个重要而古老的问题。想象一下，取一个表面(例如气球的表面)并在它上刺一个洞；空穴，即缺失点，称为穿孔，在过去的十年左右，有一些特殊的技术被发展来研究被穿孔表面S上几何结构的空间T(S)。值得注意的是，粒子物理的弦理论中的各种量是由这些空间T(S)对于被穿孔表面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的见解，而本项目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. ***