Title: Recent progress in stochastic topology

Abstract: (This talk is intended for a broad audience, and no
particular prerequisite knowledge about topology or probability will
be assumed.) The study of random topological spaces: manifolds,
simplicial complexes, knots, and groups, has received a lot of
attention in recent years. This talk will mostly focus on random
simplicial complexes, and especially on a certain kind of topological
phase transition, where the probability that that a certain homology
group is trivial passes from 0 to 1 within a narrow window. The
archetypal result in this area is the Erdős–Rényi theorem, which
characterizes the threshold edge probability where the random graph
becomes connected. One recent breakthrough has been in the application
of Garland’s method, which allows one to prove homology-vanishing
theorems by showing that certain Laplacians have large spectral gaps.
This reduces certain problems in stochastic topology to understanding
eigenvalues of certain random matrices, and the method has been
surprisingly successful. This is joint work with Christopher Hoffman
and Elliot Paquette.