Talks will start in the early afternoon of Tuesday, Jully 11, and end in the late morning of Thursday, July 13.
The list of the proposed topics is available here.
All the talks will be held at the institut, in the conference room of the first floor.
|13h - 14h||Greeting|
|14h - 15h30||Introduction||Sylvain Douteau|
|15h30 - 15h45||Coffee break|
|15h45 - 17h15||Various types of persistence complexes||Ghita Lebbar|
|9h - 10h||Barcodes||Adélie Garin|
|10h - 10h30||Coffee break|
|10h30 - 11h30||Stability of barcodes||Adélie Garin|
|11h30 - 13h||Lunch|
|13h - 14h30||Applications: Algorithms and computations||Daniel Robert-Nicoud|
|14h30 - 16h||Applications: Shape reconstruction||Elise Raphael|
|16h - 16h30||Coffee break|
|16h30 - 17h30||Applications: natural images and other examples||Daniel Robert-Nicoud|
|9h - 10h00||Categorical Persistent Homology||Timothy Hosgood|
|10h00 - 10h15||Coffee break|
|10h15 - 11h45||Link between persistent homology and spectral sequences||Najib Idrissi|
In this introduction, I will go over the basics of simplicial complexes, their associated chain complexes and homology. Then, I will try to give some intuition about what persistent homology is, and how to define it.
In the majority of scientific research, the analysis of results is based on data which are visualized as a set of points in the space. These points can be represented as a point cloud data which are usually large in size, in number and are unorganized. Therefore, abstract simplicial complex is an applicable feature to represent the topological data of this type of data. Various methods allow to represent these simplicial complexes. In this session, we will focus on specific methods including: Čech complexes, Vietoris-Rips complexes, Witness complexes and Delaunay triangulations. As a second part, an explanation will be carried out to develop some of the advantages of each type of complexes. Ambiguities are observed which are important to highlight.
A persistence diagram is a tool used to represent persistent homology. I will begin with the case of a one variable real function and its associated persistence diagram. Then I will give a first definition of persistence diagrams and explain why it completely describes persistent homology. Finally I'll give a more general definition using a classical algebra theorem to define persistence diagrams for a wider class of real valued functions.
Persistence diagrams describe persistent homology and have the important property of being stable under small perturbations. I will introduce the Bottleneck distance and Hausdorff distance between persistence diagrams and then state the main stability theorem and give an idea of the proof with some lemmas and inequalities, before explaining how the conditions can be weakened and give some applications.
While persistent homology is an instrument of theoretical importance on its own, one must not forget that the reason it was introduced, as well as its main application, is the topological analysis of data. As such, one needs to be able to do explicit computations in this domain in an efficient way, as the data clouds one is interested to analyze are more often than not composed by at least thousand of points, possibly living in very high-dimensional spaces. It is immediately obvious that efficient computations in this scale transcend the human ability and trespass into the domain of computers and algorithms. The goal of this talk is to present some basic algorithms and methods one can use to apply persistent homology to the study of data.
We present an application of persistent homology to the local study of natural images following Carlsson et al.
After explaining the general setting of shape reconstruction, I will define the Delaunay triangulation and its restricted form in order to present some of the current methods in the literature.I will then focus on the algorithm in [GO] : definition of the witness complex, 2d and 3d cases and advantages of this new method.
Using the language of categories, many of the definitions of persistent homology
and its stability theorems become simpler to read. Further, we can generalise to
extended persistent homology (where we ensure that all classes die within finite
time, thus ensuring that all critical points are paired) using the same language,
and formulate an analogous stability theorem.
This talk will cover the necessary definitions from category theory (categories, functors, natural transformations, and diagrams).
Bubenik, Peter, and Jonathan Scott. 2014. “Categorification of Persistent Homology.” arXiv math.AT (1205.3669v3).
Cohen-Steiner, David, Herbert Edelsbrunner, and John Harer. 2007. “Stability of Persistence Diagrams.” Discrete & Computational Geometry 37 (1): 103–20.
———. 2009. “Extending Persistence Using Poincaré and Lefschetz Duality.” Foundations of Computational Mathematics 9 (1): 79–103..
Dans cet exposé, j'expliquerai le lien entre l'homologie persistante et les suites spectrales. Plus précisément, après des rappels sur les suites spectrales, je montrerai comment calculer les rangs des groupes d'homologie persistante d'un complexe filtré à partir des rangs des pages de la suite spectrale associée à la filtration et vice-versa.