Topology and group theory

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Groups are fundamental mathematical objects that encode symmetries; for instance, those of a crystalline material. They arise in many fields including geometry, arithmetic, and even mathematical physics. They provide a bridge between algebraic and geometric properties of a mathematical object. The principal investigator will study problems that traverse this bridge in both directions, applying algebra to uncover geometric properties of low-dimensional spaces and using geometry to shed light on problems that appear to be strictly algebraic. An important theme is the concept of cohomology, which roughly speaking measures high-dimensional holes that exist in both geometric and algebraic settings.The principal investigator will study the topology of arithmetic groups, the mapping class group, and the automorphism groups of free groups. The planned research has two parts. In the first, the investigator will study duality properties of these groups, which will be used to perform a series of cohomological calculations. In the second part, the investigator will study the topology and geometry of the Torelli group. Foci of these projects include finiteness properties of the Torelli group, the theory of representation stability, and relationships with invariants of homology 3-spheres.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.