Homology-vanishing Theorems for Random Simplicial Complexes

Wednesday, March 12, 2014 - 1:30pm - 2:30pm
Lind 305
Matthew Kahle (The Ohio State University)
Linial-Meshulam random 2-complexes are analogues of Erdős-Rényi random
graphs, and their topological properties have been the subject of
extensive study. We now know a few methods for proving homology-vanishing
theorems in settings like this: (1) cocycle counting, i.e. combinatorial
methods, (2) spectral methods, using concentration results for spectral
gaps of certain random matrices, and (3) what might be called random
linear algebra. Each of these has its advantages, and I will briefly
overview each. The main new result I'll discuss by the end of the talk is
using the third method to describe the threshold for vanishing of homology
with integer coefficients, a problem which was very resistant to earlier
methods. This is joint work with Christopher Hoffman and Elliot Paquette.