Universality in Random Topology

Random geometric complexes are simplicial complexes (high-dimensional graphs) whose vertices are generated by a random point process in a metric space. In this talk we will focus on the homology (cycles/holes in various dimensions) of these complexes. Our main results show that the lifetime distribution of homological cycles obeys a universal law, that depends on neither the support nor the original distribution of the point process. We will focus on the notion of “weak universality”, addressing Poisson or binomial processes. We will present the main universality statement and the key steps for proving it. In fact, we will show that this notion of universality applies in a much broader context to scale-invariant geometric functionals (for example, the degree distribution in the k-NN graph). In addition, we will briefly discuss “strong universality”, which applies for a much wider class of point-cloud distributions, and is currently an open conjecture.