Le calcul tensoriel sur les variétés différentielles comprend l'arithmétique des champs tensoriels, le produit tensoriel, les contractions, la symétrisation et l'antisymétrisation, la dérivée de Lie le long d'un champ vectoriel, le transport par une application différentiable (pullback et pushforward), mais aussi les opérations intrinsèques aux formes différentielles (produit intérieur, produit extérieur et dérivée extérieure). On ajoutera également toutes les opérations sur les variétés pseudo-riemanniennes (variétés dotées d'un tenseur métrique) : connexion de Levi-Civita, courbure, géodésiques, isomorphismes musicaux et dualité de Hodge.Dans ce cours, nous introduirons tout d'abord la problématique du calcul tensoriel formel, en distinguant le calcul dit “abstrait” du calcul explicite. C'est ce dernier qui nous intéresse ici. Il se ramène in fine au calcul symbolique sur les composantes des champs tensoriels dans un champ de repères, ces composantes étant exprimées en termes des coordonnées d'une carte donnée.
Nous discuterons alors d'une méthode de calcul tensoriel générale, valable sur l'intégralité d'une variété donnée, sans que l'utilisateur ait à préciser dans quels champs de repères et avec quelles cartes doit s'effectuer le calcul. Cela suppose que la variété soit couverte par un atlas minimal, défini carte par carte par l'utilisateur, et soit décomposée en parties parallélisables, i.e. en ouverts couverts par un champ de repères. Ces contraintes étant satisfaites, un nombre arbitraire de cartes et de champs de repères peuvent être introduits, pourvu qu'ils soient accompagnés des fonctions de transition correspondantes.
Nous décrirons l'implémentation concrète de cette méthode dans SageMath ; elle utilise fortement la structure de dictionnaire du langage Python, ainsi que le schéma parent/élément de SageMath et le modèle de coercition associé. La méthode est indépendante du moteur de calcul formel utilisé pour l'expression symbolique des composantes tensorielles dans une carte. Nous présenterons la mise en œuvre via deux moteurs de calcul formel différents : Pynac/Maxima (le défaut dans SageMath) et SymPy. Différents champs d'application seront discutés, notamment la relativité générale et ses extensions.[-]

Le calcul tensoriel sur les variétés différentielles comprend l'arithmétique des champs tensoriels, le produit tensoriel, les contractions, la symétrisation et l'antisymétrisation, la dérivée de Lie le long d'un champ vectoriel, le transport par une application différentiable (pullback et pushforward), mais aussi les opérations intrinsèques aux formes différentielles (produit intérieur, produit extérieur et dérivée extérieure). On ajoutera également toutes les opérations sur les variétés pseudo-riemanniennes (variétés dotées d'un tenseur métrique) : connexion de Levi-Civita, courbure, géodésiques, isomorphismes musicaux et dualité de Hodge.Dans ce cours, nous introduirons tout d'abord la problématique du calcul tensoriel formel, en distinguant le calcul dit “abstrait” du calcul explicite. C'est ce dernier qui nous intéresse ici. Il se ramène in fine au calcul symbolique sur les composantes des champs tensoriels dans un champ de repères, ces composantes étant exprimées en termes des coordonnées d'une carte donnée.
Nous discuterons alors d'une méthode de calcul tensoriel générale, valable sur l'intégralité d'une variété donnée, sans que l'utilisateur ait à préciser dans quels champs de repères et avec quelles cartes doit s'effectuer le calcul. Cela suppose que la variété soit couverte par un atlas minimal, défini carte par carte par l'utilisateur, et soit décomposée en parties parallélisables, i.e. en ouverts couverts par un champ de repères. Ces contraintes étant satisfaites, un nombre arbitraire de cartes et de champs de repères peuvent être introduits, pourvu qu'ils soient accompagnés des fonctions de transition correspondantes.
Nous décrirons l'implémentation concrète de cette méthode dans SageMath ; elle utilise fortement la structure de dictionnaire du langage Python, ainsi que le schéma parent/élément de SageMath et le modèle de coercition associé. La méthode est indépendante du moteur de calcul formel utilisé pour l'expression symbolique des composantes tensorielles dans une carte. Nous présenterons la mise en œuvre via deux moteurs de calcul formel différents : Pynac/Maxima (le défaut dans SageMath) et SymPy. Différents champs d'application seront discutés, notamment la relativité générale et ses extensions.[-]

The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral geometry. The most well known example is Smale's 17th problem, which asks to find a solution of a given system of n complex homogeneous polynomial equations in $n$ + 1 unknowns. This problem can be solved in average (and even smoothed) polynomial time.
In the course we will explain the concepts necessary to state and solve Smale's 17th problem. We also show how these ideas lead to new numerical algorithms for computing eigenpairs of matrices that provably run in average polynomial time. Making these algorithms more efficient or adapting them to structured settings are challenging and rewarding research problems. We intend to address some of these issues at the end of the course.[-]

The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral geometry. The most well known example is Smale's 17th problem, which asks to find a solution of a given system of n complex homogeneous polynomial equations in $n$ + 1 unknowns. This problem can be solved in average (and even smoothed) polynomial time.
In the course we will explain the concepts necessary to state and solve Smale's 17th problem. We also show how these ideas lead to new numerical algorithms for computing eigenpairs of matrices that provably run in average polynomial time. Making these algorithms more efficient or adapting them to structured settings are challenging and rewarding research problems. We intend to address some of these issues at the end of the course.[-]

Structure is a fundamental concept in linear algebra: matrices arising from applications often inherit a special form from the original problem, and this special form can be analysed and exploited to design efficient algorithms. In this short course we will present some examples of matrix structure and related applications. Here we are interested in data-sparse structure, that is, structure that allows us to represent an n × n matrix using only O(n) parameters. One notable example is provided by quasi separable matrices, a class of (generally dense) rank-structured matrices where off-diagonal blocks have low rank.
We will give an overview of the properties of these structured classes and present a few examples of how algorithms that perform basic tasks – e.g., solving linear systems, computing eigenvalues, approximating matrix functions – can be tailored to specific structures.[-]

Algebraic combinatorics studies combinatorial objects with an algebraic point of view, and conversely. As such, it is also a very fertile ground for experimental mathematics, involving both classical and new algorithms. I will discuss two topics: finite partially ordered sets and their invariants, and tree-indexed polynomials and power series. Finite partially ordered sets are discrete objects, that can be seen as directed graphs, but also possess an interesting representation theory. This leads to many difficult questions about a subtle equivalence relation, namely posets having equivalent derived categories. The theme of tree-indexed series, which can be traced back to Cayley, plays a role in the study of vector fields and ordinary differential equations. It is nowadays better understood in the framework of operads and can be considered as a nonassociative version of the study of alphabets, words and languages. Surprisingly maybe, rooted trees also appear in the study of iterated integrals, stemming out of the usual "integration-by-part" rule. I will describe the corresponding notions of algebras, without diving too much into the theory of operads. On the way, I will discuss some of the involved algorithms and their implementations.[-]

I will start from reviewing Gröbner bases and their connection to polynomial system solving. The problem of solving a polynomial system of equations over a finite field has relevant applications to cryptography and coding theory. For many of these applications, being able to estimate the complexity of computing a Gröbner basis is crucial. With these applications in mind, I will review linear-algebra based algorithms, which are currently the most efficient algorithms available to compute Gröbner bases. I will define and compare several invariants, that were introduced with the goal of providing an estimate on the complexity of computing a Gröbner basis, including the solving degree, the degree of regularity, and the last fall degree. Concrete examples will complement the theoretical discussion.[-]

Robotic design involves modeling the behavior of a robot mechanism when it moves along potential paths set by the users. In this lecture, we will first give an overview of different approaches to design a set of kinematic equations associated with a robot mechanism. In particular, these equations can be used to solve the forward and the backward kinematics problems associated with a robot mechanism or to model its singularity locus. Then we will review methods to solve those equations, and notably methods to draw with guarantees the real solutions of an under-constrained system of equations modeling the singularities of a robot.[-]

Structure is a fundamental concept in linear algebra: matrices arising from applications often inherit a special form from the original problem, and this special form can be analysed and exploited to design efficient algorithms. In this short course we will present some examples of matrix structure and related applications. Here we are interested in data-sparse structure, that is, structure that allows us to represent an n × n matrix using only O(n) parameters. One notable example is provided by quasi separable matrices, a class of (generally dense) rank-structured matrices where off-diagonal blocks have low rank.
We will give an overview of the properties of these structured classes and present a few examples of how algorithms that perform basic tasks - e.g., solving linear systems, computing eigenvalues, approximating matrix functions - can be tailored to specific structures.[-]

Algebraic combinatorics studies combinatorial objects with an algebraic point of view, and conversely. As such, it is also a very fertile ground for experimental mathematics, involving both classical and new algorithms. I will discuss two topics: finite partially ordered sets and their invariants, and tree-indexed polynomials and power series. Finite partially ordered sets are discrete objects, that can be seen as directed graphs, but also possess an interesting representation theory. This leads to many difficult questions about a subtle equivalence relation, namely posets having equivalent derived categories. The theme of tree-indexed series, which can be traced back to Cayley, plays a role in the study of vector fields and ordinary differential equations. It is nowadays better understood in the framework of operads and can be considered as a nonassociative version of the study of alphabets, words and languages. Surprisingly maybe, rooted trees also appear in the study of iterated integrals, stemming out of the usual "integration-by-part" rule. I will describe the corresponding notions of algebras, without diving too much into the theory of operads. On the way, I will discuss some of the involved algorithms and their implementations.[-]