Szego Seminar: Instantons and the 4-color theorem

Abstract:  To a trivalent graph in 3-space, Kronheimer and Mrowka showed how to associate a certain finite dimensional Z/2 vector space, called the instanton homology. Using techniques from 3-manifold topology, they prove that this invariant is nonzero whenever the graph satisfies some reasonable conditions. It is conjectured that for planar graphs the rank of this vector space is equal to the number of Tait colorings of the graph, which are essentially the same as 4-colorings of the regions enclosed by the graph. If this conjecture were to hold, it would partner with the nonvanishing theorem to give a proof of the 4-color theorem which, unlike the existing proofs, does not reply on computer verification. We'll give some background discussion and then motivate how gauge theoretic methods come in to play.