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C04
Persistence and Stability of Geometric Complexes


The following more detailed questions are studied within this project: The definition and construction of geometric complexes from data. Topological persistence. The homology of dynamical systems. The convergence of variants of Crofton's formula obtained with persistent homology to compute intrinsic volumes. The approximation of persistent homology through simplification of the representative complexes.

Mission-

The investigation of stable topological features of geometric complexes and the development of computational methods to leverage this topological information for applications in dynamical systems and data analysis.

Scientific Details+

The project is concerned with multiscale families of complexes constructed from geometric or abstract data. Commonly, the data is a sample of some shape (a subspace of some metric space), and the complexes serve, in a precise sense, as a discretization of the topology of the shape at a given scale. The complexes in the family are linked by maps, and the resulting diagram contains global information that reveals more structure than just the individual complexes appearing in the diagram.  

We will investigate stable topological features of these complexes and develop computational methods to leverage this topological information for applications in dynamics and data analysis. One particular motivating problem arises from the study of invariant sets of discrete dynamical systems. 

Utilizing tools of discrete Morse theory, we aim to provide an efficient computational link between different construction of geometric complexes, and between different complexes approximating the same shape. An application of these methods lies in the problem of computing the homology of a self-map from a sample, a relevant problem in the study of discrete dynamical systems using Conley index theory.  

We have new results on the expected number of critical simplices and intervals of the radius functions of the Poisson--Delaunay mosaic in n-dimensional space.  Specifically, we have the expected numbers in dimensions n ≤ 4, and we their distribution dependent on the radius.  The expected number of Delaunay simplices in dimensions n ≤ 4 follow. 

The stability of persistence diagrams has recently been used to obtain integral geometric formulas for the first intrinsic volume (in R3 the total mean curvature) that converge for cubical approximations of shapes with smooth boundary.  The extension to most other intrinsic volumes is still open.  We propose to study possible extensions, in particular to the second intrinsic volume (in R3 the area). 

Publications+

Papers
Persistence in sampled dynamical systems faster

Authors: Bauer, Ulrich and Edelsbrunner, Herbert and Jablonski, Grzegorz and Mrozek, Marian
Note: Preprint
Date: Sep 2017
Download: arXiv

Expected sizes of Poisson--Delaunay mosaics and their discrete Morse functions

Authors: Edelsbrunner, Herbert and Nikitenko, Anton and Reitzner, Matthias
Journal: Advances in Applied Probability, 49(3):745--767
Date: 2017
Download: arXiv

The Morse theory of Čech and Delaunay complexes

Authors: Bauer, Ulrich and Edelsbrunner, Herbert
Journal: Transactions of the American Mathematical Society, 369(5):3741--3762
Date: 2017
DOI: 10.1090/tran/6991
Download: external arXiv

Persistence Diagrams as Diagrams: A Categorification of the Stability Theorem

Authors: Bauer, Ulrich and Lesnick, Michael
Note: Preprint
Date: Nov 2016
Download: arXiv

Algebraic stability of zigzag persistence modules

Authors: Botnan, Magnus Bakke and Lesnick, Michael
Note: Preprint
Date: 2016
Download: arXiv


Team+

Prof. Dr. Ulrich Bauer   +

Projects: C04
University: TU München, Geometrie & Visualisierung (M10)
Address: Boltzmannstraße 3 D-85747, Garching
E-Mail: ulrich.bauer[at]ma.tum.de
Website: https://www.professoren.tum.de/bauer-ulrich/

Prof. Dr. Herbert Edelsbrunner   +

Projects: C04
University: Institute of Science and Technology Austria
E-Mail: edels[at]ist.ac.at
Website: https://ist.ac.at/research/research-groups/edelsbrunner-group/

Dr. Magnus Botnan   +

Projects: C04
University: TU München
E-Mail: botnan[at]ma.tum.de

Benedikt Fluhr   +

Projects: C04
University: TU München
E-Mail: fluhr[at]ma.tum.de

Dr. Grzegorz Jablonski   +

Projects: C04
University: Institute of Science and Technology Austria
E-Mail: grzegorz.jablonski[at]ist.ac.at

Fabian Lenzen   +

Projects: C04
University: TU München
E-Mail: fabian.lenzen[at]tum.de

Anton Nikitenko   +

Projects: C04
University: Institute of Science and Technology Austria
E-Mail: anton.nikitenko[at]ist.ac.at

Dr. Florian Pausinger   +

Projects: C04
University: TU München

Dr. Hubert Wagner   +

Projects: C04
University: Institute of Science and Technology Austria
E-Mail: hubert.wagner[at]ist.ac.at