Abstract: In this talk, we shall state central limit theorems for many local and global functionals of simplicial complexes built on various random point processes. In the first part of the talk we will consider simplicial counts in Cech and Voronoi simplicial complexes for long-range dependent point processes such as zeros of Gaussian analytic functions and determinantal point processes. These functionals serve as a good illustration of our general central limit theorems for local functionals of the above point processes.
In the second part, we shall restrict ourselves to the ubiquitous Poisson point process but look at a very global functional called the Betti number. We shall show various stabilizing properties of the Betti numbers of the random Cech complex to leverage recent results on stabilizing functionals of Poisson point processes.