Algebra and topology of the Johnson filtration

为了更深入地理解映射类群的代数结构，PI将研究Torelli群，它是辛表示的核心。更一般地说，PI将研究Johnson过滤，它是映射类群的子群序列，从Torelli群开始。一个主要的悬而未决的问题是Torelli群是否被有限地呈现。Pi还想了解伪阿诺索夫膨胀是如何分布在约翰逊过滤项中的。最后，PI的目标是找到约翰逊过滤和相关组项的发电机组。映射类组描述了表面的对称性，或者类似于我们生活的三维空间的二维物体。物体的对称集通常最容易用矩阵来理解。在映射类组的情况下，有一种自然的方法可以使用矩阵来理解图像的大部分，但是也有大量的信息没有被这些矩阵检测到。PI正在研究的正是映射类组的这个神秘部分。

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.