Mini-course: Topology of Random Abstract Simplicial Complexes

Matthew Kahle, Ohio State University.

Abstract: The Erdos-Renyi random graph is a model of fundamental importance in combinatorics, probability, and statistical mechanics.
In recent years several researchers have studied various higher-dimensional generalizations of this kind of random graph, and especially their topological properties. One important theme in this area is the idea of a "phase transition", for example the homology-vanishing threshold, where the probability that homology is trivial passes from 0 to 1 within a very narrow window.

In this mini-course, we will overview this area. I will assume that the audience is familiar with the basic definitions of topology: homology, fundamental group, etc. But otherwise the course will be self contained, and will begin by reviewing various topological features of random graphs before studying their higher-dimensional analogues.