Group trisections and smooth four-manifolds

In this talk, I will introduce the notion of a trisection of a group, and I'll explain how studying group trisections is equivalent to studying smooth structures on 4-manifolds.

The proof of this striking theorem, which is the main result of a recent eponymous paper by Abrams, Gay, and Kirby, involves a straightforward and beautiful synthesis of foundational theorems from the theories of 2-, 3-, and 4-dimensional manifolds. After presenting the proof, I will conclude by discussing connections to the Poincare Conjecture, Andrews-Curtis Conjecture, and knotted surface theory.