topology@ub

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

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.

• Guest Talks (Spring 2019)
• Presentable infinity categories (Fall 2018 and Spring 2019)
• Guest Talks (Spring 2018)
• Homotopy types for Khovanov homology (Spring 2018)
• Fall Seminar 2017 (Fall 2017)
• Reading group on ∞-categories (Fall 2016)
• Postnikov towers and localization (Spring 2015)
• Spectral sequences (Fall 2014)
• Higher categories (Spring 2013)
• Noncommutative motives (Fall 2011)
• Spring Seminar 2011 (Spring 2011)
• Deformation theory (Fall 2010)
• Summer Seminar 2010 (Summer 2010)
• DG-categories (Spring 2010)
• Topological quantum field theories (Fall 2009)
• Quantum field theories (Fall 2009)