Title: PhD thesis defense: torsion in homology of random simplicial complexes
Speaker: Andrew Newman (Ohio State University)
Abstract: During the mid-twentieth century, Paul Erdős and Alfréd Rényi developed their now-standard random graph model. Beyond being practical in graph theory to non- constructively prove the existence of graphs with certain interesting properties, the Erdos–Rényi model is also a model for generating random (one-dimensional) topological spaces. Within the last fifteen years, this model has been generalized to the higher-dimensional simplicial complex model of Nati Linial and Roy Meshulam. As in the case of the probabilistic method more generally, there are (at least) two reasons why one might apply random methods in topology: to understand what a "typical" topological space looks like and to give nonconstructive proofs of the existence of topological spaces with certain properties. Here we consider both of these applications of randomness in topology in considering the properties of torsion in homology of simplicial complexes. For the former, we discuss experimental results that strongly suggest torsion in homology of random Linial–Meshulam complexes is distributed according to Cohen Lenstra heuristics. For the latter, we use the probabilistic method to give an upper bound on the number of vertices required to construct d-dimensional simplicial complexes with prescribed torsion in homology. This upper bound is optimal in the sense that it is a constant multiple of a known lower bound.