GATMAID EMS Summer School

The central topic of the school is Topologically Data Analysis, its impact in other areas of research and its emerging applications in real life problems. The school is organized in three courses of 6/7 hours each: the first one introduces the basic framework, while the other two address specific topics in Algebra and in Geometry that are currently growing out of the impulse of TDA. Each course will have two parts, one presenting the mathematical background and another one showing its potential for applications. Participants will be given information in advance on the recommended background for the school and during the school there will be tutorial/discussion hours.

The scientific part of the school is complemented with:

– few research talks by senior speakers, to illustrate some specific aspects related with them

– a poster session, in which willing participants will be able to present their work

– an activity open to general public. Consisting on a talk on ethic aspects related to Artificial Intelligence, and a round table with title: From pure research to technological transfer: a new era for mathematics?

The courses will be suitable for early PhD students or even Master level students willing to learn on the topic and that may be looking for prospective PhD topics and advisors. It will be a truly international event as speakers and participants will be coming from across the globe (although mostly from Europe) and the summer school is a prelude to the Conference LIGAT.

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 class 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

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:

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.

Differential calculus for modules over posets

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.

Antonio Viruel

Universidad de Málaga

This activity is an initiative of the  research group LIGAT-Laboratori d’Interaccions entre Geometria, Àlgebra i Topologia (2021 SGR 01015). We also thank the funding of 

We are also grateful to Red Temática de Álgebra no Conmutativa (RED2022-134631-T) for its collaboration.

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