Algebra and topology of the Johnson filtration

为了更深入地了解映射类群的代数结构，PI将研究作为辛表示的核心的Torelli群。更一般地，PI将调查Johnson过滤，它是从Torelli群开始的映射类群的子群的序列。一个主要的悬而未决的问题是，Torelli群是否被有限地表示。PI还想了解伪阿诺索夫扩张是如何在约翰逊过滤的术语中分配的。最后，PI的目标是找到Johnson滤子及其相关群的生成集，映射类群描述曲面或类似于我们生活的三维空间的二维对象的对称性。通常通过使用矩阵来理解对象的对称性集是最容易的。在映射类组的情况下，有一种自然的方法来使用矩阵来理解画面的一大部分，但也有大量信息逃脱了这些矩阵的检测。PI正在研究的正是映射班级群体中的这一神秘部分。

In order to more deeply understand the algebraic structure of the mapping class groups, the PI will study the Torelli group, which is the kernel of the symplectic representation. More generally, the PI will investigate the Johnson filtration, which is a sequence of subgroups of the mapping class group starting with the Torelli group. A major open question is whether or not Torelli groups are finitely presented or not. The Pi would also like to understand how pseudo-Anosov dilatations are distributed among the terms of the Johnson filtration. Finally, the PI aims to find generating sets for terms of the Johnson filtration and related groups.The mapping class group describes the symmetries of surfaces, or two-dimensional objects which are analogous to the three-dimensional space we live in. The set of symmetries of an object is often most easily understood by using matrices. In the case of the mapping class group, there is a natural way to use matrices to understand a large piece of the picture, but there is also a large amount of information that escapes detection by these matrices. It is this mysterious part of the mapping class groups that the PI is studying.