Gauge Theory and Trivalent Graphs in Three-Manifolds

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

In this research project, the principal investigator will continue work that aims to provide a human-readable proof of the four color map theorem. The four color map theorem was first proved in 1976 by Appel and Haken; the proof involved a huge amount of case checking, so much so that the only feasible method of attack was via computer. This machine-assisted proof, however, cannot be directly checked by a human. Although there have been some subsequent simplifications and clarifications, the general brute-force approach has seemed to be the only line of attack. The four color theorem can be rephrased, though, as a question about the topology of three-dimensional manifolds, and in particular new tools inspired by high energy physics give novel ways to approach a conceptual proof of the four color theorem. This project explores and develops these new insights. The principal investigator and collaborator define a version of instanton Floer homology for trivalent graphs embedded in three-manifolds. This theory enables them to place the four color map theorem firmly in the context of gauge-theory-inspired invariants of three-manifolds. They have proved a fundamental non-vanishing theorem for this Floer homology group, reducing the four color map theorem to a question about computing this invariant for general planar graphs. They are developing new tools for the computation of these and related invariants, and they have discovered a spectral sequence relevant to this computation. The collapse of the spectral sequence for planar graphs would imply the four color theorem. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.