Braid groups and related structures

The braid group is a fundamental and classical object arising in both algebraic and geometric contexts. This project will explore the various actions of the braid group on spaces with particularly nice properties, in order to learn more about its group structure. One important open question is whether braid groups act on CAT(0) spaces, that is, spaces with nonnegative curvature. In the past 15 years, this question has been settled for the cases of braids up to 7 strands, and one direction of this project will look at mechanisms for extending this to braids of arbitrarily many strands. Another direction is to look at actions of the braid groups on linear spaces, e.g., to look at representations of the braid group, including its symplectic representations, and to investigate algebraic properties of their kernels, such as their abelianizations.