Surface graphs, gain sparsity and some applications in discrete geometry

Abstract: A collection of line segments in the plane forms a 2-contact system if the segments have pairwise disjoint interiors and no pair of segments have an endpoint in common. Thomassen has shown that a graph is the intersection graph of such a 2-contact system if and only if it is a subgraph of a planar Laman graph. Also Haas, Orden, Rote, Francisco, Servatius, Servatius, Souvain, Streinu and Whiteley have shown that a graph admits a plane embedding as a pointed pseudotriangulation if and only if is a planar Laman graph. I will discuss recent work on symmetric versions of these results. In this context the graphs that arise are naturally embedded in the orbifold associated to the action of the symmetry group, and the appropriate sparsity conditions are gain sparsity conditions. Our main tools are new topological inductive constructions for the appropriate classes of surface graphs. All of the work presented here is joint with Bernd Schulze.

combinatorics

Audience: researchers in the topic

( slides | video )

Virtual seminar on algebraic matroids and rigidity theory

Series comments: The COVID-19 pandemic is forcing us all to stay home, foregoing conferences and departmental seminars for the next few months. This weekly virtual seminar is an attempt to patch that departmental-seminar-sized void in our lives until it is safe to resume our more traditional forms of professional networking. Since geographic location matters a lot less for a virtual seminar than for an in-person seminar, this virtual seminar will be defined purely by its mathematical theme, algebraic matroids and rigidity theory, and not any particular department nor region.