Given n random points on a manifold embedded in a Euclidean space, we wish to understand what topological features of the manifold can be inferred from the geometry of these points. One procedure is to consider union of Euclidean balls of radius r around the points and study the topology of this random set (i.e., the union of balls). A more robust method (known as persistent homology) of inferring the topology of the underlying manifold is to study the evolution of topology of the random set as r varies. What topological information (for example, Betti numbers and some generalizations) about the underlying manifold can we glean for different choices of r ? This question along with some partial answers in the recent years will be the focus of the talk. I shall try to keep the talk mostly self-contained assuming only knowledge of basic probability and point-set topology.

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Last updated: 09 Aug 2024