Mukherjee

Stochastic topology

Abstract:I will talk about a few results in stochastic topology.

  • Given n points drawn from a uniform distribution on a manifold we place balls of size r around each point. The random object we consider is the union of the balls. As n goes to infinity and r goes to zero we consider the limiting distribution of topological summaries such as Betti numbers and critical points. We provide scaling limits for this process.
  • We introduce the persistent homology transform (PHT), to model surfaces and shapes. We use the PHT to represent shapes and execute operations such as computing distances between shapes or classifying shapes. We show that the map from the space of simplicial complexes into the space spanned by the PHT is injective. This implies that we can use it to determine a metric on the space of piecewise linear shapes.  We apply the PHT to study primate heel bones.
  • We introduce a consistent estimator for the homology of level sets of both density and regression functions.