Date & Time: Monday, March 03, 2014, 16:00-17:00.
Venue: Ramanujan Hall
Title: Central limit theorems for some random simplicial complexes
Speaker: D. Yogeshwaran, Technion
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. Apart from proving normal convergence, for certain regimes, we shall be able to give 'optimal' rates of convergences as well. 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. Time permitting, we shall hint at strong laws or Poisson limit theorems for some of these functionals.
The necessary notions of simplicial complexes and point processes shall be defined in the talk. The talk is based on recent work with R.J.Adler, E.Subag, B.Blaszczyszyn and J.E.Yukich.