# Discrete CATS Seminar

Title: BOGO Sale

Abstract: This talk will consist of two 20 minute talks which I will be giving at the Joint Math Meetings. Feedback will be solicited.

2:00 - 2:20 Title: k-matching sequences of simplicial complexes.

Abstract: The homotopy type of the matching complex, M_1(G), has been studied for paths, cycles, and trees. In this talk we will generalize 1-matching complex to k-matching complexes, denoted M_k(G) and consider the sequence (M_1(G), M_2(G),…,M_n(G)) up to homotopy for perfect caterpillar graphs.

2:25 - 2:45 Title: A positivity phenomenon in Elser’s Gaussian-cluster percolation model

Abstract: Veit Elser proposed a random graph percolation model in which physical dimension appears as a parameter. From this model, numerical graph invariants els_k(G) , called Elser numbers, naturally arise and Viet Elser conjectured that els_k(G) \geq 0 for all graph $G$ and integers k \geq 2. In this talk we will interpret the Elser numbers as Euler characteristics of nucleus (simplicial) complexes and prove Elser’s conjecture. This is joint work with Galen Dorpalen-Barry, Cyrus Hettle, David Livingston, Jeremy Martin, George Nasr, and Hays Whitlatch.