Algebra and topology of the Johnson filtration

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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.