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将研究约翰逊过滤，这是从Torelli组开始的映射类组的一系列子组。 一个主要的悬而未决的问题是Torelli群是否是可证明的。 Pi还想了解伪Anosov迭代如何分布在约翰逊过滤的项中。最后，PI的目的是找到生成集的约翰逊过滤和相关的groups.The映射类组描述的对称性的表面，或二维物体类似于我们生活在三维空间。 一个物体的对称性集合通常用矩阵最容易理解。 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一大片图片，但也有大量的信息逃脱了这些矩阵的检测。 PI正在研究的正是映射类组的这一神秘部分。

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.