This project concerns the mapping class group, which is the group of isotopy classes of homeomorphisms of a surface. There is particular emphasis on the Torelli group, the subgroup consisting of elements that act trivially on the homology of the surface. We take a three- pronged approach, on the homological, dynamical, and group-theoretical properties of mapping class groups. The techniques include methods from combinatorial Morse theory, Teichmuller theory, and geometric group theory. In joint work with Bestvina and Bux, we aim to determine the homological finiteness properties of the Torelli groups; in particular to answer the famous question of whether or not they are finitely presented. With Farb and Leininger, we work on understanding the structure of all low entropy pseudo-Anosovs; in particular to address the Symmetry Conjecture, which asserts that all such pseudo- Anosovs are obtained as the product of a rotation with a map supported on a subsurface of uniformly bounded genus. With Brendle, we study the group structure of the hyperelliptic Torelli group, in particular focused on the conjecture of Hain, which states that the group is generated by Dehn twists.
The focus of this project is the mapping class group, which is the collection of symmetries of a surface, such as a sphere or a torus. This is a central object in mathematics, with connections to algebraic geometry, hyperbolic geometry, and dynamics. Discoveries in this area have had influence on fields outside of mathematics, such as string theory and cryptography. With my collaborators, I have made progress on decades-old questions about the global structure of mapping class group, and this project is dedicated to answering some of the most important remaining questions. The techniques involved in this study are wide-ranging, from combinatorial, to algebraic, to topological, to dynamical, and the proposed work has applications to algebraic geometry, 3-manifolds, and group theory. I will work on the research aspects of the proposal with several collaborators and with students at various levels. In addition, I will host a workshop for young researchers in topology, with the aim of training and cultivating future researchers and educators.
