Mathematical Sciences: The Moduli Space of Curves, its Level-n Covers and Torelli Space

9401611本提案涉及对三个相关专题的研究。第一个是主要研究者在计算曲线模空间的同调方面的工作的继续。提出了一种新的技术，它涉及已知的模空间三角剖分和Arbarello旗帜之间的相互作用。第二个主题是把关于模空间的结果推广到它的n阶覆盖和Torelli空间。第三个是与David Epstein和Steve Kerckhoff的合作，他们正在寻找自由群的外自同构群的自动结构。这项研究是一些数学家和物理学家正在进行的理解数学的努力的一部分，数学是理论物理学最近某些发展的基础。特别是，引力是自然界四种基本作用力中唯一没有合理量子理论的一种(量子理论描述物质的小行为，而普通理论描述大的行为)。物理学家(主要是Witten)提出了一种量子引力的方法，但理解它所需的数学涉及曲线的模数空间，目前还没有得到充分的理解。***

9401611 Harer This proposal concerns research on three related topics. The first is a continuation of the principal investigator's work on the computation of the homology of the moduli space of curves. New techniques are proposed which involve the interplay between the known triangulations of moduli spaces and the Arbarello flag. The second topic is the generalization of the results on moduli space to its level-n covers and to Torelli space. The third is joint work with David Epstein and Steve Kerckhoff in which they are looking for automatic structures for the outer automorphism group of a free group. This research is part of an ongoing attempt by a number of mathematicians and physicists to understand the mathematics which underlies certain recent developments in theoretical physics. In particular, gravity is the only one of the four fundamental forces of nature for which there is no reasonable quantum theory (quantum theories describe the behavior of matter in the small, while ordinary theories describe behavior in the large). An approach to quantum gravity was suggested by the physicists (primarily Witten), but the mathematics necessary to understand it involves the moduli space of curves and is not yet understood sufficiently. ***