CAREER: The topology of infinite groups

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The investigator will study the topology of the mapping class group and arithmetic groups. The proposed research has three parts. In the first, the investigator will use representation theory to study the stable cohomology of these groups and their subgroups. A special focus will be on congruence subgroups and the Torelli group. In the second part, the investigator will study vanishing conjectures for the high-dimensional unstable cohomology of these groups. Finally, in the third part the investigator will study the topology and geometry of the Torelli subgroup of the mapping class group. Foci of these projects include the structure of the Johnson filtration of the Torelli group and relationships with the Casson invariant of homology 3-spheres.Groups are mathematical objects that encode symmetries. The groups discussed in this proposal are related to the symmetries of two fundamental types of objects. The mapping class group encodes the symmetries of two-dimensional surfaces like the surface of a donut or the earth. Arithmetic groups encode symmetries of special kinds of structures that arise when studying natural numbers (i.e. 1, 2, 3,...). For mysterious reasons, these groups share many fundamental properties. The investigator will study these shared properties with a focus on the "cohomology" of the groups, which in some sense is a measure of higher-dimensional "holes" that exist in them. The investigator will also run a mathematical summer program for disadvantaged high-school students from the Houston public schools.