Mathematical Sciences: Geometry and Toplogy of Riemann's Space

9322042研究人员彭纳将继续研究黎曼曲面的模空间的几何和拓扑。具体地说，他将研究Teichmueller空间的一个自然单纯完备化，其中伴随的模空间的紧化被猜想为orborold；这种紧化应该是几何中几个计数问题的有用工具。我们将研究模空间的各种算术方面：多对数恒等式应该从模空间上的典范形式得到；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Grothendieck‘dessin d’fant图相关的数域应该为研究者现有的胖图数据库计算。这个数据库应该用来在计算机上计算各种模空间的同调群。此外，关于他的泛Teichmueller空间，还有许多未解决的分析和算术问题需要研究，该空间就是模为Moebius群的圆的保定向同胚空间。上面的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，被称为“黎曼曲面的模空间”，已经被研究了几个世纪。粗略地说，这个模空间是固定的二维曲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，也是因为高能物理的新的和令人兴奋的接口。事实上，许多理想化但有趣的真实物理问题相当于关于模空间本身的几何的问题，这提供了大量的技术、问题和想法，无论是数学还是物理。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，并将她的某些研究与高能物理相结合。***

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. ***