Conference abstracts

Session A4 - Computational Geometry and Topology

July 11, 15:00 ~ 15:25 - Room B7

Intrinsic $1$-dimensional persistence of geodesic spaces

Ziga Virk

IST Austria, Austria   -

Given a nice compact geodesic space $X$, such as a compact Riemannian manifold, we present the theory of intrinsic $1$-dimensional persistence, i.e., persistence on all points of $X$ induced by the geodesic distance. While the induced complexes have uncountably many vertices, the mentioned persistence turns out to be pretty tame and has very precise geometric interpretation. It also gives a specific understanding on how the size of holes is measured by persistence.

It turns out that persistence via Rips or Cech construction contains precisely the same information in both cases. This information represents the collection of lengths of the 'shortest' generating set. Furthermore, critical triangles can be used to localize minimal representatives of homology classes, and the approximation by finite samples is much more stable than suggested by standard stability theorems.

In the end we will outline how one may extract one-dimensional geometric information about $X$ from certain higher-dimensional persistence features.

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FoCM 2017, based on a nodethirtythree design.