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

小行星9401611 这项建议涉及三个相关专题的研究。 首先是延续的主要研究者的工作计算的同源性的模空间的曲线。 提出了新的技术，其中涉及的模空间和Arbarello标志之间的相互作用的已知三角剖分。 第二个主题是模空间的结果推广到它的n层覆盖和Torelli空间。 第三个是联合工作与大卫爱泼斯坦和史蒂夫Kerckhoff在他们正在寻找自动结构的外部自同构群的自由组。 这项研究是一些数学家和物理学家正在进行的尝试的一部分，以了解理论物理学某些最新发展的数学基础。 特别是，引力是自然界四种基本力中唯一没有合理量子理论的力（量子理论描述的是物质的微观行为，而普通理论描述的是宏观行为）。 物理学家（主要是维滕）提出了一种量子引力的方法，但理解它所必需的数学涉及曲线的模空间，而且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. ***