### Recent progress in stochastic topology

- Series
- School of Mathematics Colloquium
- Time
- Thursday, November 12, 2015 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Prof. Dr. Matthew Kahle – Ohio State University – mkahle@math.osu.edu

The study of random topological spaces: manifolds,
simplicial complexes, knots, and groups, has received a lot of
attention in recent years. This talk will 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
problems in random 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.