A way to prove the Four Color Theorem using gauge theory

In this talk, I show how ideas coming out of gauge theory can be used to prove that certain configurations in the list of “633 unavoidable’s” are reducible. In particular, I show how to prove the most important initial example, the Birkhoff diamond (four “adjacent” pentagons), is reducible using our filtered $3$- and $4$-color homology. In this context reducible means that the Birkhoff diamond cannot show up as a “tangle” in a minimal counterexample to the 4CT. This is a new proof of a 111-year-old result that is a direct consequence of a special (2+1)-dimensional TQFT. I will then indicate how the ideas used in the proof might be used to reduce the unavoidable set of 633 configurations to a much smaller set.

This is joint work with Ben McCarty.