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CRM News Institutional María de Maeztu

Los Sesgos Inconscientes: Reflexiones de la catedrática Capitolina Díaz

20/06/2024
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“Mates i a xalar” a la ràdio: xerrades de matemàtiques divulgatives a les Terres de l’Ebre

18/06/2024
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Hypatia 2024

12/06/2024
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El CRM participa un any més a la Festa de la Ciència

11/06/2024
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New Scientific Research Sheds Light on Ecological Dynamics and Late Warning Signals in Conservation Efforts

23/05/2024
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The CRM participates in the MSCA-COFUND TOUCH Project

23/05/2024
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Eva Miranda, Inclosa a la Iniciativa #científicasCASIO

13/05/2024
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El CRM celebra quaranta anys com a referent internacional de les matemàtiques

10/05/2024
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Exploring Career Pathways for Mathematics Graduates: Insights from the ‘And after the Degree, What?’ Roundtable

22/04/2024
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18/04/2024
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04/04/2024
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Participació del CRM a l’EspaiCiència

03/04/2024
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02/04/2024
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Estudiants de l’Institut La Vall del Tenes de Barcelona visiten el CRM

25/03/2024
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Exposició sobre terratrèmols i tsunamis amb una mirada matemàtica a la biblioteca de ciències de la UAB

19/03/2024
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18/03/2024
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La Agencia Estatal de Investigación financia 564 proyectos de matemáticas que mejorarán la seguridad y la salud de la población

14/03/2024
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11/03/2024
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29/02/2024
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selected videos

Centre de Recerca Matemàtica

Centre de Recerca Matemàtica
COURSE 2: Algebraic structures in Topological Data Analysis: Persistent homology Lecturers: Thomas Brüstle | Université de Sherbrooke and Bishop’s University Course Description: Topological data analysis (TDA) uses topology to identify relevant geometric features of data, such as clusters and loops. This is accomplished by first modelling the data by a family of topological spaces indexed over a poset, and then identifying the topological features that persist over several indices. The field of algebraic topology provides tools to TDA via the language of homology, and persistent homology is the branch of TDA that makes use of methods from representation theory in order to study these representations of posets, referred to as persistence modules. In this course we will present the basic theory of pointwise finite persistence modules with coefficients in a field. This includes classification results for one-parameter persistence modules where the poset is totally ordered, as well as description of invariants of persistence modules in the multiparameter setting. We will also discuss some notions that are useful in the applications, such as the stability of an invariant, which, roughly speaking means that two persistence modules that come from similar data should have close values of their corresponding invariants. We will also show how the theory of persistence modules helps in practice in the study of the shape of data. Course Materials: Preliminary Readings: + Steve Oudot: Persistence Theory: From Quiver representations to Data Analysis, https://geometrica.saclay.inria.fr/team/Steve.Oudot/books/o-pt-fqrtda-15/surv-209.pdf. Chapter 1 + Magnus Bakke Botnan and Michael Lesnick: An Introduction to Multiparameter Persistence https://arxiv.org/pdf/2203.14289
Algebraic structures in Topological Data Analysis: Persistent homology | 1/5
The Geometry of Data: Curvature and Beyond Lecturers: Emil Saucan | Braude College of Engineering Course Description: The geometrization of data, after its timid first steps, has become a mainstream tool in Data Analysis. Among the geometric notions residing at the core of this approach, curvature naturally plays a prominent role. We therefore first explore the various notions of network curvatures, their mathematical motivations, relationships, relative advantages and disadvantages and their applications to network intelligence: classification, sampling and beyond. Extensive experiments and case studies will be included. Another dominant mathematical tool applied in Data Analysis is that of Persistent Homology. Moreover, there is a connection between it and curvature. Indeed, it was observed experimentally that Persistent Homology of networks and hypernetworks schemes based on Forman’s discrete Morse Theory on the one hand, and on the 1-dimensional version of Forman’s Ricci curvature on the other, not only perform well, but they also produce practically identical results. We show that this apparently paradoxical fact can be easily explained in terms of Banchoff’s discrete Morse Theory. This allows us to prove that there exists a curvature-based, efficient Persistent Homology scheme for networks and hypernetworks. Moreover, we show that the proposed method can be broadened to include more general types of networks, by using Bloch’s extension of Banchoff’s work and even made more efficient by making appeal to quite recent results of Grunert, Kühnel and Rote. We conclude with an exploration of the power in networks classification and understanding of other types of (geo-)metric invariants, introduced in the theoretical setting by Grove and Markvorsen. Their efficiency is illustrated by experiments with complex networks and images.
The Geometry of Data: Curvature and Beyond | 3/5
COURSE 1: Topological Data Analysis Lecturers: Frédéric Chazal | Inria Course Description: Topological Data Analysis is a sound family of techniques that is gaining an increasing importance for the interactive analysis and visualization of data in imaging and machine learning applications. Given the increasing complexity and size of current collections of acquired or simulated data-sets (2D, 3D and nD), these approaches aim at helping users understand the complexity of their data by providing insights about its topological and geometric structure. In low dimensions (typically 2 or 3), Topological Data Analysis enables users to rapidly extract, interact with, and classify geometric features defined by level sets or integral lines. Thanks to simplification mechanisms based on Persistent Homology, such algorithms additionally construct multi-scale topological representations of the data, that enable users to perform robust analyses and comparisons despite the presence of noise. The soundness, efficiency and robustness of this of approaches made it increasingly popular in the last few years in a variety of 2D and 3D imaging analysis applications. In higher dimensions, these techniques have recently been adapted to form the basis of new clustering algorithms and data analysis tools. The purpose of this course is to introduce the main concepts of the recent field of Topological Data Analysis and illustrate their use in imaging (scientific visualization) and machine learning applications, both from a mathematical and practical point of view. Course Materials: Preliminary Readings: + The following chapters of the book “J.D. Boissonnat, F. Chazal, M. Yvinec. Geometric and topological Inference. Cambridge Univ. Press” would be good as preliminary readings (a free pdf of the book is available here: https://inria.hal.science/hal-01615863/ ) : Notions of geometric and abstract simplicial complexes : 2.1 – 2.5 Basic notions of simplicial homology and singular homology : 11.1 – 11.3 + A recommend general overview (not necessarily a preliminary reading): https://www.frontiersin.org/articles/10.3389/frai.2021.667963/full
Topological Data Analysis | 3/4
The Geometry of Data: Curvature and Beyond Lecturers: Emil Saucan | Braude College of Engineering Course Description: The geometrization of data, after its timid first steps, has become a mainstream tool in Data Analysis. Among the geometric notions residing at the core of this approach, curvature naturally plays a prominent role. We therefore first explore the various notions of network curvatures, their mathematical motivations, relationships, relative advantages and disadvantages and their applications to network intelligence: classification, sampling and beyond. Extensive experiments and case studies will be included. Another dominant mathematical tool applied in Data Analysis is that of Persistent Homology. Moreover, there is a connection between it and curvature. Indeed, it was observed experimentally that Persistent Homology of networks and hypernetworks schemes based on Forman’s discrete Morse Theory on the one hand, and on the 1-dimensional version of Forman’s Ricci curvature on the other, not only perform well, but they also produce practically identical results. We show that this apparently paradoxical fact can be easily explained in terms of Banchoff’s discrete Morse Theory. This allows us to prove that there exists a curvature-based, efficient Persistent Homology scheme for networks and hypernetworks. Moreover, we show that the proposed method can be broadened to include more general types of networks, by using Bloch’s extension of Banchoff’s work and even made more efficient by making appeal to quite recent results of Grunert, Kühnel and Rote. We conclude with an exploration of the power in networks classification and understanding of other types of (geo-)metric invariants, introduced in the theoretical setting by Grove and Markvorsen. Their efficiency is illustrated by experiments with complex networks and images.
The Geometry of Data: Curvature and Beyond | 2/5
The Geometry of Data: Curvature and Beyond Lecturers: Emil Saucan | Braude College of Engineering Course Description: The geometrization of data, after its timid first steps, has become a mainstream tool in Data Analysis. Among the geometric notions residing at the core of this approach, curvature naturally plays a prominent role. We therefore first explore the various notions of network curvatures, their mathematical motivations, relationships, relative advantages and disadvantages and their applications to network intelligence: classification, sampling and beyond. Extensive experiments and case studies will be included. Another dominant mathematical tool applied in Data Analysis is that of Persistent Homology. Moreover, there is a connection between it and curvature. Indeed, it was observed experimentally that Persistent Homology of networks and hypernetworks schemes based on Forman’s discrete Morse Theory on the one hand, and on the 1-dimensional version of Forman’s Ricci curvature on the other, not only perform well, but they also produce practically identical results. We show that this apparently paradoxical fact can be easily explained in terms of Banchoff’s discrete Morse Theory. This allows us to prove that there exists a curvature-based, efficient Persistent Homology scheme for networks and hypernetworks. Moreover, we show that the proposed method can be broadened to include more general types of networks, by using Bloch’s extension of Banchoff’s work and even made more efficient by making appeal to quite recent results of Grunert, Kühnel and Rote. We conclude with an exploration of the power in networks classification and understanding of other types of (geo-)metric invariants, introduced in the theoretical setting by Grove and Markvorsen. Their efficiency is illustrated by experiments with complex networks and images.
The Geometry of Data: Curvature and Beyond | 1/5
COURSE 1: Topological Data Analysis Lecturers: Frédéric Chazal | Inria Course Description: Topological Data Analysis is a sound family of techniques that is gaining an increasing importance for the interactive analysis and visualization of data in imaging and machine learning applications. Given the increasing complexity and size of current collections of acquired or simulated data-sets (2D, 3D and nD), these approaches aim at helping users understand the complexity of their data by providing insights about its topological and geometric structure. In low dimensions (typically 2 or 3), Topological Data Analysis enables users to rapidly extract, interact with, and classify geometric features defined by level sets or integral lines. Thanks to simplification mechanisms based on Persistent Homology, such algorithms additionally construct multi-scale topological representations of the data, that enable users to perform robust analyses and comparisons despite the presence of noise. The soundness, efficiency and robustness of this of approaches made it increasingly popular in the last few years in a variety of 2D and 3D imaging analysis applications. In higher dimensions, these techniques have recently been adapted to form the basis of new clustering algorithms and data analysis tools. The purpose of this course is to introduce the main concepts of the recent field of Topological Data Analysis and illustrate their use in imaging (scientific visualization) and machine learning applications, both from a mathematical and practical point of view. Course Materials: Preliminary Readings: + The following chapters of the book “J.D. Boissonnat, F. Chazal, M. Yvinec. Geometric and topological Inference. Cambridge Univ. Press” would be good as preliminary readings (a free pdf of the book is available here: https://inria.hal.science/hal-01615863/ ) : Notions of geometric and abstract simplicial complexes : 2.1 – 2.5 Basic notions of simplicial homology and singular homology : 11.1 – 11.3 + A recommend general overview (not necessarily a preliminary reading): https://www.frontiersin.org/articles/10.3389/frai.2021.667963/full
Topological Data Analysis | 2/4

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