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.
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.
- Group: C. Computation
- Principal Investigators: Prof. Dr. Ulrich Bauer, Prof. Dr. Herbert Edelsbrunner
- Investigators: Dr. Magnus Botnan, Benedikt Fluhr, Dr. Grzegorz Jablonski, Fabian Lenzen, Anton Nikitenko, Dr. Florian Pausinger, Dr. Hubert Wagner
- Universities: TU München, Institute of Science and Technology Austria
- Term: since 2016
Mission-
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
Algebraic Stability of Zigzag Persistence Modules
Authors:
Botnan, Magnus Bakke and
Lesnick, Michael
Journal: Algebraic & Geometric Topology
Date:
Dec 2018
Download:
external
Using persistent homology to reveal hidden covariates in systems governed by the kinetic Ising model
Authors:
Spreemann, Gard and
Dunn, Benjamin and
Botnan, Magnus Bakke and
Baas, Nils A.
Journal: Journal: Phys. Rev. E 97, 032313
Date:
Mar 2018
DOI:
10.1103/PhysRevE.97.032313
Download:
external
Hardness of Approximation for Morse Matching
Authors:
Bauer, Ulrich and
Rathod, Abhishek
Journal: preprint
Date:
Jan 2018
Download:
arXiv
Persistent Betti numbers of random Čech complexes
Authors:
Bauer, Ulrich and
Pausinger, Florian
Journal: preprint
Date:
Jan 2018
Download:
arXiv
The Reeb Graph Edit Distance is Universal
Authors:
Bauer, Ulrich and
Landi, Claudia and
Memoli, Facundo
Journal: preprint
Date:
Jan 2018
Download:
arXiv
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
Interval decomposition of infinite zigzag persistence modules
Author:
Botnan, Magnus Bakke
Journal: Proc. Amer. Math. Soc. 145 (2017), 3571-3577
Date:
Jan 2017
DOI:
10.1090/proc/13465
Download:
external
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
Computational Complexity of the Interleaving Distance
Authors:
Bjerkevik, Havard Bakke and
Botnan, Magnus Bakke
Journal: Proceedings of the 34th International Symposium on Computational Geometry (SoCG 2018)
Date:
DOI:
10.4230/LIPIcs.SoCG.2018.13
Download:
external
Posters
The Mayer-Vietoris Pyramid Sheaf-theoretically
Author:
Fluhr, Benedikt
Journal: Poster Session: Bridging Statistics and Sheaves at IMA, Minneapolis
Date:
May 2018
Download:
external
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