Introduction to braid groups

Part 1 (14:00-15:00): Introduction to braid groups

Braid groups are groups that are defined by figures formed by the entanglement of n strings. Besides this geometric realization, it is a very interesting field where algebra and analysis intersect. In the first half of this seminar, aimed mainly at those unfamiliar with braid groups, we will introduce three aspects of braid groups and review the history of the research. In particular, in the area of its relation to analysis, the relationship between KZ equations and braid groups will be introduced.

Part 2 (15:30-16:30): Representations of braid groups and the relationship between monodromy representations of KZ equations

In the second half of the talk, after a brief introduction to representation theory, we will introduce the Katz-Long-Moody construction, a method of constructing infinite series of representations of the semi-direct product of braid group and free group. We will also show that its special case is isomorphic to multiplicative middle convolution, a method for constructing monodromy representations of KZ equations. Lastly, we will also discuss the connection between representations of braid groups and knot invariants. The talk includes joint work with Kazuki Hiroe.