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Workshop

A Sheaf Theoretic Construction of Shape Space

  • Shreya Arya (Duke University)
E1 05 (Leibniz-Saal)

Abstract

In this talk I will present a sheaf theoretic construction of shape space (or the space of all shapes). We do this by describing a homotopy sheaf on the poset category of constructible sets, where each set is mapped to its Persistent Homology Transform (PHT). Recent results that build on fundamental work of Schapira have shown that this transform is injective, thus making the PHT a good summary object for each shape. Our homotopy sheaf result allows us to "glue" PHTs of different shapes together to build up the PHT of a larger shape. In the case where our shape is a polyhedron we prove a generalized nerve lemma for the PHT. Finally, by re-examining the sampling result of Smale-Niyogi-Weinberger, we show that we can reliably approximate the PHT of a manifold by a polyhedron up to arbitrary precision. This is joint work with Justin Curry and Sayan Mukherjee.

Katharina Matschke

Max Planck Institute for Mathematics in the Sciences, Leipzig Contact via Mail

Karen Habermann

University of Warwick

Sayan Mukherjee

Max Planck Institute for Mathematics in the Sciences, Leipzig

Max von Renesse

Leipzig University

Stefan Horst Sommer

University of Copenhagen