On the topology of some random complexes

There has been recent interest in understanding the homology of random simplicial complexes built over point processes. I shall start with some basic definitions in algebraic topology and then try to describe results about homology of Cech and Vietoris-Rips complexes built over random point-sets. Both these complexes have random point-sets on the Euclidean space as vertices and the faces are determined by some deterministic geometric rule. The aim of the talk shall be to explain the quantitative differences in the growth of homology groups measured via Betti numbers between different point processes. I shall also try to hint at the proof techniques which involve detailed analysis of subgraph and component counts of the associated random geometric graphs and are applicable to similar functionals of point processes such as Morse critical points. This is a joint work with Prof. Robert Adler.

For most of the talk only basic knowledge of probability and topology shall be required.