This talk was given remotely via Zoom and was projected in MLT 206.

Most fundamental questions about random graphs are related to their connectivity properties. Such properties often exhibit phase transitions - sharp change in behavior as a response to a small perturbation to the model parameters. In this talk we will focus on two such phase transitions: (1) percolation - when a “giant” connected component emerges, (2) connectivity - when the entire graph becomes connected. We will introduce and discuss higher-dimensional analogues for these phenomena that occur in random simplicial complexes. Our generalized notions for connectivity and percolation will be using the language of algebraic topology, and specifically homology theory. The main focus will be on geometric simplicial complexes. These complexes are generated by taking the vertices to be a random point process, and adding simplexes according to their local geometric configuration. We will discuss some recent results related to these topological phase transitions.