Topology Seminar (Main talk): Combinatorics of Legendrian Surfaces
Seminar | February 22 | 2:10-3 p.m. | 736 Evans Hall
Eric Zaslow, Northwestern University
I want to describe an approach to Legendrian surface theory using cubic planar graphs. After describing how to associated a Legendrian surface to such a graph, I will give a purely combinatorial description of the moduli space of objects in a sheaf category equivalent to the Fukaya category of Lagrangian threefolds in complex three-space with asymptotics defined by the Legendrian surface. The moduli space is the space of map colorings of the triangulation dual to the graph, with colors chosen in the projective line, modulo the diagonal action of fractional linear transformations. The chromatic polynomial is of the graph is then an isotopy invariant of the corresponding Legendrian surface.
Depending on time and on the interests of the audience, I could also explain the connection to open Gromov-Witten theory, to cluster theory, or describe attempts to generalize the story to cubic spectral networks.
This talk is based on joint works with Treumann, Treumann-Shen, and on work in progress.