Research Group in Algebraic Topology

Current Seminar

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

    [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.

    Past Seminars