We'll talk about joint work with Jacob Lurie regarding moduli stacks of geometric objects developing natural breaks. If time allows, I'll end with some speculation regarding a 3-D TFT arising from various G2 manifolds.

The famous Burau representation of the braid group is known to be unfaithful for braids with at least five strands. In the early 2000s, two constructions were provided to fix faithfulness: the first being the Lawrence–Krammer–Bigelow linear representation, hence proving linearity of braid groups, and the second being the Khovanov–Seidel categorical representation. In this talk, based on joint work in progress with Licata, Queffelec, and Wagner, I will investigate the interplay between these two representations.