Abstract: |
I will begin by explaining how flops arise in algebraic geometry, and how it is difficult to iterate. However, when viewed from a cluster/homological perspective, it becomes easier, and I will outline what structures we expect to see algebraically. The expectation, now proved in many cases, is that flops induce some type of "affine pure braid group" acting on the derived category. I will explain the Coxeter combinatorics background behind this, which is completely independent of the geometric motivation. The remarkable thing is that the geometry predicts unseen algebraic phenomenon, like an affine version of the symmetries of the pentagon (which does not exist!), and most of the talk will be motivated by the simplest case of two intersecting flopping curves. Towards the end I will explain how this case is related to certain tilings of the plane, and indeed every choice of two nodes of a Dynkin diagram induces a tiling of the plane. These tilings seem to be new, and are somewhat mysterious. |