Geometric uses of persistent homology
The purpose of this seminar is to learn some of the recent advances on the interaction between persistent homology theory
and geometry, mostly in the setting of symplectic and algebraic geometry.
Organizers:
Joana Cirici and Marco Praderio (Universitat de Barcelona)
-
Marco Praderio (UB):
Inferring topology through barcodes and point clouds
30 October 2019, 10:30, Aula S1
Abstract:
An introduction to persistence in homology and its depiction by means of barcodes, persistence diagrams, and landscapes, including an example from a study on digital photographs of random outdoor scenes. [ Slides ] [ Notes ] -
Joana Cirici (UB): Classification of persistent modules and some geometric examples
13 November 2019, 10:30, Aula S1
Abstract:
I will introduce persistent modules and explain how they can be classified by simple combinatorial objects called barcodes. We will also see that the space of persistent modules admits a pseudo-metric, defined via the interleaving distance, which will provide an Isometry Theorem when considering barcodes and the so-called bottleneck distance. I will try to motivate the discussion with some very first examples related to Morse theory and geometry. [ Notes ] -
Ignasi Mundet (UB):
Rigidity in symplectic geometry and persistence modules
4 December 2019, 10:30, Aula S1
Abstract:
Faré una breu introducció a la geometria simplèctica, posant èmfasi en resultats de rigidesa. Donaré una descripció impressionista de l'homologia de Floer, i explicaré com aquesta homologia permet definir mòduls de persistència que poden ser usats per entendre qüestions de rigidesa en geometria simplèctica. -
Mikel Lluvia (UB): Isometry and stability
11 December 2019, 10:30, Aula S1
Abstract:
Statement and proof of the Stability Theorem, relating the Hausdorff distance on point clouds with the bottleneck distance on barcodes. -
Carles Casacuberta (UB):
Further about stability
15 January 2020, 10:30, Aula S1
Abstract:
The Gromov-Hausdorff distance between compact metric spaces can be defined by means of distortions of correspondences or, equivalently, in terms of isometric embeddings. In the previous lecture, it was shown that the bottleneck distance between the Vietoris-Rips barcodes of two point clouds X and Y is bounded by twice the Gromov-Hausdorff distance between X and Y. In this lecture, we will use stability for smooth functions in order to discuss bounds for the bottleneck distance between Čech barcodes. [ Notes ] -
Joana Cirici (UB):
Introduction to Floer homology
29 January 2020, 10:30, Aula S1
Abstract:
Els mòduls de persistència simplèctics es defineixen a partir d'una versió filtrada de l'homologia de Floer. Aquesta teoria homològica és, en certa manera, una versió per a espais de dimensió infinita de la teoria de Morse. Hi intervé l'espai de llaços contràctils d'una varietat simplèctica, així com eines potents d'anàlisi funcional i de la teoria de corbes pseudo-holomorfes. Les aplicacions de la teoria de Floer són immenses en topologia simplèctica i topologia de dimensió baixa. En aquesta xerrada introduïrem els ingredients mínims necessaris per a donar una idea de la construcció de la homologia de Floer Hamiltoniana associada a una varietat simplèctica compacta. [ Notes ] -
Carles Casacuberta (UB):
Dynamical stability for Hamiltonian persistence modules
5 February 2020, 10:30, Aula S1
Abstract:
We will review the Hofer distance between Hamiltonian diffeomorphisms of a closed symplectic manifold, and prove that the bottleneck distance between the barcodes of two non-degenerate Hamiltonian diffeomorphisms is bounded by the Hofer distance between them. The proof uses the assumption that the second homotopy group of the manifold vanishes. The extent to which this assumption is necessary will be discussed.
References
[1] H. Edelsbrunner, J. Harer, Persistent homology: A survey, Contemp. Math., vol. 453, Amer. Math. Soc., Providence, 2008, 257-282.
[2] Y. I. Manin, M. Marcolli, Nori diagrams and persistent homology,
arXiv:1901.10301
(2019).
[3] L. Polterovich, D. Rosen, K. Samvelyan, J. Zhang, Topological persistence in geometry and analysis,
arXiv:1904.04044 (2019).
[4] S. Weinberger, Interpolation, the rudimentary geometry of spaces of Lipschitz functions, and geometric complexity, Found. Comput. Math. 19 (2019), 991-1011.