Behind the scenes with triple grid diagrams

Contact for this takl: Aru Ray (ray@mpim...)

This is a report on joint work with Sarah Blackwell and Peter Lambert-Cole. Lagrangians are half-dimensional submanifolds in symplectic manifolds on which the symplectic form vanishes; they are the even-dimensional analogues of Legendrians in contact manifolds, and in fact Lagrangians with boundary give the correct notion of cobordism between Legendrians. We are interested in Lagrangian surfaces in symplectic 4-manifolds, and in the relative setting in Lagrangian cobordisms between Legendrian knots in contact 3-manifolds. I will describe a program to understand these combinatorially and give an explicit "behind the scenes" peek in local coordinates on R^4 to help you see how to construct them and why the combinatorial setup really works. As a preview, one way to describe the combinatorial data is as an immersed trivalent graph in R^2 whose edges are all straight line segments, and such that at each vertex one edge is horizontal, one edge is vertical, and one edge has slope -1. An entertaining warmup is to draw some such graphs.