Persistence and stability of geometric complexes

Project Period: 2016-09-01 – 2020-08-31
Externally Funded
Principal Investigator
Herbert Edelsbrunner
Department(s)
Edelsbrunner Group
Description
The topic belongs to the general area of Computational Topology, and more specifically to the Persistent Homology sub-area. Building on recent results by the two authors, the project suggests several extensions and improvements of past work. A. We propose a stochastic analysis of the discrete Morse theory of the Delaunay triangulation of a Poisson point process. B. We aim at extending the proof of convergence of the modified Crofton Formula for intrinsic volumes. C. We to develop topological data analysis for Bregman divergences and analyze its stability. D. We study sparse complexes that preserve or approximate the persistence of standard distance functions. E. We apply the discrete Morse theory of Cech and Delaunay complexes to find solutions to sampled dynamical systems. Each problem will need its own methods, and solutions to one problem will feed into approaches to others. As an overarching method, we mention that discrete Morse theory -- as developed by Robin Forman about 20 years ago -- applies to all five topics and ties the work together to a coherent project. If successful, the advances will create a discrete theory that bridges several up to now disjoint mathematical topics. The completion of this theory is driven by the applications of the results and the corresponding computational tools to data analysis questions.
Grant Number
I 2979 -N35
Funding Organisation
FWF

7 Publications

2019 | Journal Article | IST-REx-ID: 6608   OA
Holes and dependences in an ordered complex
H. Edelsbrunner, K. Ölsböck, Computer Aided Geometric Design 73 (2019) 1–15.
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2018 | Conference Paper | IST-REx-ID: 187   OA
The multi-cover persistence of Euclidean balls
H. Edelsbrunner, G.F. Osang, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018.
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2017 | Journal Article | IST-REx-ID: 718
Expected sizes of poisson Delaunay mosaics and their discrete Morse functions
H. Edelsbrunner, A. Nikitenko, M. Reitzner, Advances in Applied Probability 49 (2017) 745–767.
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2018 | Journal Article | IST-REx-ID: 312
On the optimality of the FCC lattice for soft sphere packing
H. Edelsbrunner, M. Iglesias Ham, SIAM J Discrete Math 32 (2018) 750–782.
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2018 | Journal Article | IST-REx-ID: 5678
Poisson–Delaunay Mosaics of Order k
H. Edelsbrunner, A. Nikitenko, Discrete and Computational Geometry (2018).
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2018 | Conference Paper | IST-REx-ID: 188   OA
Smallest enclosing spheres and Chernoff points in Bregman geometry
H. Edelsbrunner, Z. Virk, H. Wagner, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018, p. 35:1-35:13.
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2018 | Journal Article | IST-REx-ID: 87
Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics
H. Edelsbrunner, A. Nikitenko, Annals of Applied Probability 28 (2018) 3215–3238.
View | Files available | DOI | Download (ext.) | arXiv