Topology at a Scale in Metric Spaces

Nat Smale ()
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Nat Smale: University of Utah, Department of Mathematics

A chapter in Essays in Mathematics and its Applications, 2012, pp 421-430 from Springer

Abstract: Abstract This is an expository article that discusses some developments in joint work with Laurent Bartholdi, Thomas Schick and Steve Smale in [1] and also in [10]. Recently, in various contexts, there has been interest in the topology of certain spaces (even finite data sets) at a “scale”, for example, in reconstruction of manifolds or other spaces from a discrete sample as in [8] and [4], and also in connection with learning theory [9, 11] and [7]. In persistence homology, [3, 5] mathematicians have been computing topological features at a range of scales, to find the fundamental structures of spaces and data sets. See also [2]. In this paper, we will first give an explicit description of homology at a scale, for a compact metric space. We will then describe a Hodge theory for the corresponding cohomology when the space has a Borel probability measure.

Keywords: Riemannian Manifold; Simplicial Complex; Quotient Space; Borel Probability Measure; Hodge Theory (search for similar items in EconPapers)
Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-28821-0_16

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DOI: 10.1007/978-3-642-28821-0_16

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