Topology and motives associated to moduli spaces of curves

这个项目的主要重点是霍奇和伽罗瓦结构的映射类组的完成和它们的关系，混合动机的模空间的曲线。一个特别有趣的例子是在属1中，PI正在研究与椭圆曲线相关的混合动机。与Makoto松本（广岛大学）的PI正在研究行动的绝对伽罗瓦组的Malcev完成的基本组的一次穿孔椭圆曲线，也对相对完成相应的映射类组。与格雷戈里Pearlstein（密歇根州）的PI正在研究的变化混合霍奇结构的模空间的椭圆曲线和它们的关系，迭代积分的模块化形式最近定义的尤里马宁。伽罗瓦和霍奇结构是平行的，PI希望每种方法都能照亮另一种。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制。在伽罗瓦方面，松本和PI正在证明关于算术映射类群完备化的伽罗瓦动作的基本结果。在霍奇方面，Pearlstein、Terasoma和PI正在开发基本的数学工具，用于研究复杂代数簇上混合霍奇结构的一般变化类。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刻联系。 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起来。PI与他的合作者和学生一起，正在研究这些对称群与“椭圆曲线”（由三次多项式定义的曲线）相关的相互作用。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），是一个既定的专家在这些主题。

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.