Topology and motives associated to moduli spaces of curves

这个项目的主要焦点是关于映射类群的完备性以及它们与曲线的模空间上的混合动机的关系的Hodge和Galois结构。一个特别令人感兴趣的案例是亏格1，PI正在研究与椭圆曲线有关的混合动机。PI正在与广岛大学的松本真本一起研究绝对伽罗瓦群在一条曾经被穿孔的椭圆曲线的基本群的Malcev完备性上的作用，以及在相应映射类群的相对完备性上的作用。PI与Gregory Pearlstein(密歇根州立大学)一起研究了椭圆曲线模空间上混合Hodge结构的变化，以及它们与Yuri Manin最近定义的模形式的迭代积分的关系。伽罗瓦和霍奇的结构是平行的，PI希望每种方法都能照亮另一种方法。在其他项目中，PI和他的合作者正在开发研究椭圆动机情况所需的通用机械。在Galois方面，MatSumoto和PI证明了关于算术映射类群完备化的Galois作用的基本结果。在Hodge方面，Pearlstein、Terasoma和PI正在开发基本的数学工具，用于研究复代数簇上混合Hodge结构的一般变分类。动机编码了整数理论、某些代数函数的积分和拓扑之间的深层次联系。这些理论中的每一个都有自己的一套对称性，并且都通过动机理论联系在一起。PI与他的合作者和学生一起，正在研究这些对称群之间的相互作用，这些对称群与“椭圆曲线”有关，“椭圆曲线”是由三次多项式定义的曲线。虽然这项工作是基础性的，但它在密码学和伪随机数生成方面有潜在的应用。事实上，PI的主要合作者松本(广岛大学)在这些主题上是一位公认的专家。

The main focus of this project is Hodge and Galois structures on completions of mapping class groups and their relationship to mixed motives over moduli spaces of curves. One case of particular interest is in genus 1, where the PI is studying mixed motives associated to elliptic curves. With Makoto Matsumoto (Hiroshima University) the PI is studying the action of the absolute Galois group on the Malcev completion of the fundamental group of a once-punctured elliptic curve and also on the relative completion of the corresponding mapping class group. With Gregory Pearlstein (Michigan State) the PI is studying variations of mixed Hodge structure over moduli spaces of elliptic curves and their relationship to iterated integrals of modular forms recently defined by Yuri Manin. The Galois and Hodge structures are parallel and the PI expects each approach to illuminate the other. In other projects, the PI and his collaborators are developing general machinery needed to study the case of elliptic motives. On the Galois side, Matsumoto and the PI are proving basic results about Galois actions on completions of arithmetic mapping class groups. On the Hodge side, Pearlstein, Terasoma and the PI are developing fundamental mathematical tools for studying general classes of variations of mixed Hodge structure over complex algebraic varieties.Motives encode deep connections between the theory of whole numbers, integrals of certain algebraic functions and topology. Each of these theories has its own set of symmetries, and all are related through the theory of motives. The PI, together with his collaborators and students, are investigating the interactions of these symmetry groups that are associated to ``elliptic curves'', which are curves defined by cubic polynomials. Although this work is foundational, it has potential applications to cryptography and pseudo random number generation. Indeed, the PI's principal collaborator, Matsumoto (Hiroshima University), is an established expert in these subjects.