From Gromov-Witten Theory to the Four Color Theorem

Abstract: Gromov-Witten theory is the study of pseudoholomorphic curves, i.e., maps of genus $g$ Riemann surfaces with $n$ marked points into a symplectic manifold $Y$. If the manifold $Y$ is also foliated generically by special Lagrangian tori, i.e., the SYZ conjecture, then one can study the moduli space of pseudoholomorphic maps of genus $g$ Riemann surfaces with measured foliations into $Y$ that preserve the foliations.

Riemann surfaces with measured foliations have long been known to correspond to metric ribbon graphs, i.e., special CW structures of a surface where marked points correspond to $2$-cells and each edge of the graph has a positive number associated to it (the metric). The moduli space of genus $g$ Riemann surfaces with measured foliations is a well-behaved orbifold whose points are generically given by trivalent ribbon graphs with $n$ faces.

Motivated by this background we ask: For foliated spheres with $n$ marked points, can the marked points ($2$-cells) in GW Theory be painted with four colors so that no two "adjacent" marked points have the same color? In this talk, we generate vector spaces from diagrams (that should be reminiscent of Khovanov homology) of a ribbon graph and define a differential between them based on a Frobenius algebra. We show that the dimension of the kernel of this differential is equal to the number of ways to four-face color the graph (the Four Color Theorem). We then generalize this calculation to a homology theory based upon a topological quantum field theory. The diagrams generated from the graph represent the possible quantum states of the CW structure of the sphere and the homology is, in some sense, the vacuum expectation value of this system. It gets wickedly complicated from this point on, but I hope to leave you wondering: Is the four color theorem just an extremely-difficult-to-prove oddity in graph theory, or is it tied in some fundamental way to the deeper laws of nature and space?

Believe it or not, this talk will be hands-on and the ideas will be explained through the calculation of easy examples! My goal is to attract students and mathematicians to this area by making the ideas as intuitive as possible.

mathematical physicsalgebraic geometrygeometric topology

Audience: researchers in the topic

---

Richmond Geometry Meeting 2024

Series comments: The Richmond Geometry Meeting will focus on emergent research topics while bringing together researchers in algebraic geometry, low-dimensional topology, and mathematical physics. In summer 2024, we will highlight developments in Geometric Topology and Moduli.

Zoom meeting id: 835 6496 8395 password: RGMVCU2024

---