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Threshold phenomena in random structures
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El CRM participa un any més a la Nit de la Recerca

30/09/2024
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27/09/2024
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15/09/2024
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10/09/2024
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18 anys de la Mari Paz Valero al CRM al costat de la Fundació Aura

02/08/2024
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CRM Researchers Are Developing a New Efficient Data Compression Method

01/08/2024
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30/07/2024
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CRM at the 9th European Congress of Mathematics

24/07/2024
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17/07/2024
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05/07/2024
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Los Sesgos Inconscientes: Reflexiones de la catedrática Capitolina Díaz

20/06/2024
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Hypatia 2024

12/06/2024
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The CRM’s 2023 Annual Report Highlights Growth and Collaboration

05/06/2024
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REF. 2024-018-P05005

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REF. 2024-019-P05006

Application Deadline: 15 Oct 2024

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selected videos

Centre de Recerca Matemàtica

Centre de Recerca Matemàtica
COURSE 3: 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 | 5/5
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/te... - 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 | 5/5
SPECIAL LECTURE: Differential calculus for modules over posets Lecturer: Ran Levi | University of Aberdeen Abstract: The motivation for this project arises from topological data analysis. The concept of a persistence module was introduced in the context of topological data analysis. In its general form one defines persistence modules as functors from an arbitrary poset (or more generally an arbitrary small category) to some abelian target category. In other words, a persistence module is simply a representation of the source category in the target abelian category. As such much research was dedicated to studying persistence modules in this context. Unsurprisingly, it turns out that when the source category is more general than a linear order, then its representation type is generally wild. In particular, keeping in mind that persistence module theory is supposed to be applicable, computability of general persistence modules is very limited. In this talk I will describe a new set of ideas for local analysis of modules over poset algebras by methods borrowed from spectral graph theory and multivariable calculus.
Differential calculus for modules over posets
COURSE 3: 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 | 4/5
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/te... - 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 | 4/5
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/te... - 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 | 3/5

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