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.
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.
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).
Herbert Edelsbrunner and Katharina Ölsböck.
Holes and dependences in an ordered complex.
Computer Aided Geometric Design, August 2019.
Ulrich Bauer, Magnus B. Botnan, Steffen Oppermann, and Johan Steen.
Cotorsion torsion triples and the representation theory of filtered hierarchical clustering.
preprint, April 2019.
Ulrich Bauer and Abhishek Rathod.
Hardness of Approximation for Morse Matching.
In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, 2663–2674. SIAM, Philadelphia, PA, 2019.
Magnus Bakke Botnan and Michael Lesnick.
Algebraic Stability of Zigzag Persistence Modules.
Algebr. Geom. Topol., Volume 18, Number 6 (2018), 3133-3204, December 2018.
URL: https://msp.org/scripts/coming.php?jpath=agt, arXiv:1604.00655v3, doi:10.2140/agt.2018.18.3133.
Havard Bakke Bjerkevik and Magnus Bakke Botnan.
Computational Complexity of the Interleaving Distance.
Proceedings of the 34th International Symposium on Computational Geometry (SoCG 2018), May 2018.
Gard Spreemann, Benjamin Dunn, Magnus Bakke Botnan, and Nils A. Baas.
Using persistent homology to reveal hidden covariates in systems governed by the kinetic Ising model.
Journal: Phys. Rev. E 97, 032313, March 2018.
Håvard Bakke Bjerkevik, Magnus Bakke Botnan, and Michael Kerber.
Computing the interleaving distance is NP-hard.
Ulrich Bauer and Florian Pausinger.
Persistent Betti numbers of random Čech complexes.
preprint, January 2018.
Herbert Edelsbrunner and Anton Nikitenko.
Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics.
Ann. Appl. Probab., 28(5):3215–3238, 2018.
Ulrich Bauer, Claudia Landi, and Facundo Memoli.
The Reeb Graph Edit Distance is Universal.
preprint, January 2018.
Ulrich Bauer, Herbert Edelsbrunner, Grzegorz Jablonski, and Marian Mrozek.
Persistence in sampled dynamical systems faster.
Preprint, September 2017.
Herbert Edelsbrunner, Anton Nikitenko, and Matthias Reitzner.
Expected sizes of Poisson–Delaunay mosaics and their discrete Morse functions.
Advances in Applied Probability, 49(3):745–767, 2017.
Ulrich Bauer and Herbert Edelsbrunner.
The Morse theory of Čech and Delaunay complexes.
Transactions of the American Mathematical Society, 369(5):3741–3762, 2017.
Ulrich Bauer and Michael Lesnick.
Persistence Diagrams as Diagrams: A Categorification of the Stability Theorem.
Preprint, November 2016.
The Mayer-Vietoris Pyramid Sheaf-theoretically.
May 2018. Poster Session: Bridging Statistics and Sheaves at IMA, Minneapolis.
Prof. Dr. Ulrich Bauer +
Prof. Dr. Herbert Edelsbrunner +
University: Institute of Science and Technology Austria