Mathematical Sciences: Geometry and Toplogy of Riemann's Space

小行星9322042 调查员将继续他的几何和拓扑的模空间的黎曼曲面的研究。 具体来说，他将研究Teichmueller空间的自然单纯完备，其中模空间的相关紧化被证明是一个轨道;这种紧化应该是几何中几个枚举问题的有用工具。 模空间的各种算术方面将进行研究：多对数身份应来自模空间上的规范形式;应该有一个从模空间到Volodin空间的映射，将模空间的上同调与代数K-理论联系起来;与胖图相关的数字字段a la Grothendieck's dessin d 'enfant应该为调查员现有的胖图数据库计算。 这个相同的数据库应该被用来在计算机上计算各种模空间的同调群。 还有，此外，许多突出的分析和算术问题要研究他的普遍Teichmueller空间，这只是空间的方向保持同胚的圆模的Moebius群。 上述几个项目与物理学的各种发展密切相关。 数学中的一个基本对象，称为“黎曼曲面的模空间”，已经研究了几个世纪。 粗略地说，这个模空间是一个固定的二维表面上所有可能的几何形状的集合。 在过去的十年左右，我们对模空间的理解有所增加，这既是因为数学的发展，也是因为与高能物理的新的和令人兴奋的接口。 实际上，许多理想化但有趣的真实的物理学问题相当于模空间本身的几何问题，这为数学和物理学提供了大量的技术、问题和思想。 在这个项目中，研究人员将继续他正在进行的模空间的数学研究，并将继续研究这些接口与高能物理。 ***

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. ***