Topology at a Scale in Metric Spaces

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.