Omer Bobrowski (Queen Mary, University of London)

Homological connecticity in random Čech complexes and poisson approximation

Abstract:

A well-known phenomenon in random graphs is the phase-transition for connectivity, proved first by Erdős–Rényi in 1959. In this talk we will discuss a high-dimensional analogue of this phenomenon known as "homological connectivity". Briefly, homology is an algebraic-topological structure describing various types of "cycles" that can be formed in high-dimensional shapes. Considering an increasing sequence of shapes, homological cycles are formed and filled in at various times. Homological connectivity is the point where the homology of such sequences stops changing, or "stabilizes". The model we study is the random Čech complex, which is a high-dimensional generalisation of the random geometric graph. We will show that there is a sequence of sharp phase transitions (for different degrees of homology). In addition, we will show that in each critical window, the obstructions to homological connectivity have a functional Poisson process limit.