## Geometry and Topology

•  Kevin Sackel, Stony Brook
•  Paint by Numbers... via Contact Geometry
•  02/06/2020
•  2:30 PM - 3:30 PM
•  C304 Wells Hall

Consider a trivalent planar graph embedded on the sphere. One can ask how many ways are there to color the faces of such a graph with $k$ colors so that no two adjacent faces are given the same color. Surprisingly, if $k$ is one more than a power of a prime, then one can write down a set of equations and simply count the number of solutions over a finite field to obtain essentially the same answer. This combinatorial correspondence is completely explicit and is a verification of the slogan "augmentations are sheaves" for certain Legendrian surfaces constructed by Treumann and Zaslow in standard contact $\mathbb{R}^5$. In this talk we will describe the contact-geometric framework for this result as well as the explicit combinatorial correspondence. If the speaker is daring enough, we may further speculate about the case of noncommutative augmentations and higher rank sheaves for these Legendrian surfaces, which we expect to go through a higher-dimensional generalization of the cross ratio for four elements of the Grassmannian $Gr(n,2n;F)$ for a given field $F$.

## Contact

Department of Mathematics
Michigan State University