Categorifying the four color theorem with applications to Gromov-Witten theory

Scott Baldridge (Louisiana State University)

Monday 5th February 16:00-17:00 Maths 311B

Abstract

The four color theorem states that each bridgeless planar graph has a proper 4-face coloring. It can be generalized to certain types of CW complexes of any closed surface for any number of colors, i.e., one looks for a coloring of the 2-cells (faces) of the complex with m colors so that whenever two 2-cells are adjacent to a 1-cell (edge), they are labeled different colors.

In this talk, I show how to categorify the m-color polynomial of a surface with a CW complex. This polynomial is based upon Roger Penrose’s seminal 1971 paper on abstract tensor systems and can be thought of as the "Jones polynomial’’ for CW complexes. The homology theory that results from this categorification is called the bigraded m-color homology and is based upon a topological quantum field theory (that will be suppressed from this talk due to time). The construction of this homology shares some similar features to the construction of Khovanov homology—it has a hypercube of states, multiplication and comultiplication maps, etc. Most importantly, the homology is the E_1 page of a spectral sequence whose E_infty page has a basis that can be identified with proper m-face colorings, that is, each successive page of the sequence provides better approximations of m-face colorings than the last. Since it can be shown that the E_1 page is never zero, it is safe to say that a non-computer-based proof of the four color theorem resides in studying this spectral sequence! (This is joint work with Ben McCarty.)

If time, I will relate this work to the study of the moduli space of stable genus g curves with n marked points. Using Strebel quadratic differentials, one can identify this moduli space with a subspace of the space of metric ribbon graphs with labeled boundary components. Proper m-face coloring in this setup is, in a sense, studying points in the space of metric ribbon graphs where similarly-colored boundaries (marked points) don’t get "too close’’ to each other. We will end with some speculations about what this might mean for Gromov-Witten theory of Calabi-Yau manifolds.

Note to students: This talk will be hands-on with ideas explained through the calculation of examples. Graduate students and researchers who are interested in graph theory, topology, or representation theory are encouraged to attend.

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