# Matthew Kahle : Homology of geometric random complexes

There has been a flurry of recent activity in studying the topology of point cloud data. However, there is a feeling that we are lacking rigorous null hypotheses to compare with the results. This is one motivation for the following: Take n points, independently and identically distributed in R^d, according to some distribution (for example, a standard normal distribution). Connect them if they are close (within distance epsilon, a function of n), and then build the Cech complex or Rips complex. What can one say about the homology of this complex as n approaches infinity? Or the persistent homology with respect to the radius? Using a variety of techniques, including Poissonization, Stein's method, and discrete Morse theory, we are able to identify phase transitions, and for certain ranges of epsilon prove central limit theorems for the Betti numbers. This is joint work with Gunnar Carlsson and Persi Diaconis.

**Category**: Probability**Duration**: 01:34:42**Date**: October 30, 2008 at 4:10 PM**Views**: 159-
**Tags:**seminar, Probability Seminar

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