Sharp vanishing thresholds for cohomology of random flag complexes

(These are the slides from Stanford Symposium in Algebraic Topology: Applications and New Directions, in honor of Gunnar Carlsson, Ralph Cohen, and Ib Madson.) Random flag complexes are a natural generalization of random graphs to higher dimensions, and since every simplicial complex is homeomorphic to a flag complex this puts a measure on a wide range of possible topologies. In this talk, I will discuss the recent proof that according to the Erdős–Rényi measure, asymptotically almost all $d$-dimensional flag complexes only have nontrivial (rational) homology in middle degree $ \lfloor d/2 \rfloor$. The highlighted technique is originally due to Garland --- what he called “p-adic curvature” in a somewhat different context. This method allows one to prove cohomology-vanishing theorems by showing that certain discrete Laplacians have sufficiently large spectral gap. This reduces certain questions in probabilistic topology to questions about random matrices. Some of this depends on new results for random matrices in joint work with Chris Hoffman and Elliot Paquette. Proving central limit theorems for Betti numbers in the non-vanishing regime was done in joint work with Elizabeth Meckes.