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

小行星9610041 修饰的Teichmueller穿孔表面理论给出了经典Teichmueller理论的一个新观点，而且这项工作与高能物理的弦理论有联系。 也有从这一丰富的几何理论的空间的莫比乌斯集团的右陪集在集团的方向保持同胚的圆（这是我们的模型的普遍Teichmueller空间）以及一个明智的普遍映射类集团，这原来是同构的（理查德）汤普森集团。 在代数数论中，这个普适理论也有不同的有趣的应用（例如马尔可夫元组、二次型的高斯积和格罗滕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。 在数学中，关于在一个固定的二维曲面S上的所有可能的几何结构的空间T（S），有一些重要而古老的问题。 想象一下，拿一个表面（比如气球的表面），在上面戳一个洞;这个洞，也就是说， 在过去的十年中，为了研究穿孔曲面S上几何结构的空间T（S），发展了一些特殊的技术。 值得注意的是，粒子物理学弦理论中的各种量都是由这些空间T（S）对穿孔面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构造的新见解，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. ***