Mathematics Seminar - 10/23/26

Abstract: Suppose a shape sitting in Euclidean space is known to us only through a finite set of sample points scattered near it. How much of the shape can be recovered, and in what sense? This is the reconstruction problem, and the standard topological tool for it is the Vietoris–Rips complex, built by declaring mutually close sample points to span a simplex. It often recovers the homotopy type of the shape, but it is an abstract complex that lives nowhere in particular. Its shadow, the union of the convex hulls of its simplices, does live in the ambient space, and so offers a genuine geometric reconstruction rather than a purely topological one.

The projection from the complex onto its shadow can be badly singular, and is understood only in low ambient dimensions. I will explain what goes wrong, and then describe a way around it: rather than fixing one scale and one sample, study the whole system of shadows as the scale shrinks and the sample fills out the shape. Borsuk's shape theory turns out to be the right setting, since it was built precisely for situations where a space is best understood through an inverse system approximating it. In the limit the projection behaves well on homotopy and homology under mild hypotheses, and in favorable cases the limit stabilizes and recovers the shape up to its embedding. Joint work with Kazuhiro Kawamura and Sushovan Majhi.