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# Research schools  | enregistrements trouvés : 96

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## Post-edited  Autour de la géométrie stochastique : polytopes aléatoires et autres modèles Calka, Pierre (Auteur de la Conférence) | CIRM (Editeur )

La géométrie stochastique est l'étude d'objets issus de la géométrie euclidienne dont le comportement relève du hasard. Si les premiers problèmes de probabilités géométriques ont été posés sous la forme de casse-têtes mathématiques, le domaine s'est considérablement développé depuis une cinquantaine d'années de part ses multiples applications, notamment en sciences expérimentales, et aussi ses liens avec l'analyse d'algorithmes géométriques. L'exposé sera centré sur la description des polytopes aléatoires qui sont construits comme enveloppes convexes d'un ensemble aléatoire de points. On s'intéressera plus particulièrement aux cas d'un nuage de points uniformes dans un corps convexe fixé ou d'un nuage de points gaussiens et on se focalisera sur l'étude asymptotique de grandeurs aléatoires associées, en particulier via des calculs de variances limites. Seront également évoqués d'autres modèles classiques de la géométrie aléatoire tels que la mosaïque de Poisson-Voronoi. La géométrie stochastique est l'étude d'objets issus de la géométrie euclidienne dont le comportement relève du hasard. Si les premiers problèmes de probabilités géométriques ont été posés sous la forme de casse-têtes mathématiques, le domaine s'est considérablement développé depuis une cinquantaine d'années de part ses multiples applications, notamment en sciences expérimentales, et aussi ses liens avec l'analyse d'algorithmes géométriques. ...

## Post-edited  On the boundary control method Oksanen, Lauri (Auteur de la Conférence) | CIRM (Editeur )

This is a survey talk about the Boundary Control method. The method originates from the work by Belishev in 1987. He developed the method to solve the inverse boundary value problem for the acoustic wave equation with an isotropic sound speed. The method has proven to be very versatile and it has been applied to various inverse problems for hyperbolic partial differential equations. We review recent results based on the method and explain how a geometric version of method works in the case of the wave equation for the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. This is a survey talk about the Boundary Control method. The method originates from the work by Belishev in 1987. He developed the method to solve the inverse boundary value problem for the acoustic wave equation with an isotropic sound speed. The method has proven to be very versatile and it has been applied to various inverse problems for hyperbolic partial differential equations. We review recent results based on the method and explain how a ...

## Post-edited  Persistence modules and Hamiltonian diffeomorphisms - Part 1 Polterovich, Leonid (Auteur de la Conférence) | CIRM (Editeur )

Theory of persistence modules is a rapidly developing field lying on the borderline between algebra, geometry and topology. It provides a very useful viewpoint at Morse theory, and at the same time is one of the cornerstones of topological data analysis. In the course I'll review foundations of this theory and focus on its applications to symplectic topology. In parts, the course is based on a recent work with Egor Shelukhin arXiv:1412.8277

## Post-edited  Introduction aux processus de fragmentation - Partie 1 Haas, Bénédicte (Auteur de la Conférence) | CIRM (Editeur )

Les processus de fragmentation sont des modèles aléatoires pour décrire l'évolution d'objets (particules, masses) sujets à des fragmentations successives au cours du temps. L'étude de tels modèles remonte à Kolmogorov, en 1941, et ils ont depuis fait l'objet de nombreuses recherches. Ceci s'explique à la fois par de multiples motivations (le champs d'applications est vaste : biologie et génétique des populations, formation de planètes, polymérisation, aérosols, industrie minière, informatique, etc.) et par la mise en place de modèles mathématiques riches et liés à d'autres domaines bien développés en Probabilités, comme les marches aléatoires branchantes, les processus de Lévy et les arbres aléatoires. L'objet de ce mini-cours est de présenter les processus de fragmentation auto-similaires, tels qu'introduits par Bertoin au début des années 2000s. Ce sont des processus markoviens, dont la dynamique est caractérisée par une propriété de branchement (différents objets évoluent indépendamment) et une propriété d'auto-similarité (un objet se fragmente à un taux proportionnel à une certaine puissance fixée de sa masse). Nous discuterons la construction de ces processus (qui incluent des modèles avec fragmentations spontanées, plus délicats à construire) et ferons un tour d'horizon de leurs principales propriétés. Les processus de fragmentation sont des modèles aléatoires pour décrire l'évolution d'objets (particules, masses) sujets à des fragmentations successives au cours du temps. L'étude de tels modèles remonte à Kolmogorov, en 1941, et ils ont depuis fait l'objet de nombreuses recherches. Ceci s'explique à la fois par de multiples motivations (le champs d'applications est vaste : biologie et génétique des populations, formation de planètes, ...

## Post-edited  School on the mathematics of strings theory : introduction Kashani-Poor, Amir-Kian (Auteur de la Conférence) | CIRM (Editeur )

This school consists of an array of courses which at first glance may seem to have little in common. The underlying structure relating gauge theory to enumerative geometry to number theory is string theory. In this short introduction, we will attempt to give a schematic overview of how the various topics covered in this school fit into this overarching framework.

## Post-edited  The local Langlands correspondence: functoriality, $L$-functions, gamma functions and the epsilon factors Prasad, Dipendra (Auteur de la Conférence) | CIRM (Editeur )

Spherical Hecke algebra, Satake transform, and an introduction to local Langlands correspondence.

## Post-edited  Applications of algebra to automatic sequences and pattern avoidance - Lecture 1 Bell, Jason P. (Auteur de la Conférence) | CIRM (Editeur )

We will cover some of the more important results from commutative and noncommutative algebra as far as applications to automatic sequences, pattern avoidance, and related areas. Well give an overview of some applications of these areas to the study of automatic and regular sequences and combinatorics on words.

## Post-edited  Characters, maps, free cumulants. Lecture 1: Characters, maps, free cumulants and Stanley character formula Sniady, Piotr (Auteur de la Conférence) | CIRM (Editeur )

- Normalized characters of the symmetric groups,
- Kerov polynomials and Kerov positivity conjecture,
- Stanley character polynomials and multirectangular coordinates of Young diagrams,
- Stanley character formula and maps,
- Jack characters
- characterization, partial results.

## Post-edited  Entropy and mixing for multidimensional shifts of finite type - Lecture 1 Pavlov, Ronnie (Auteur de la Conférence) | CIRM (Editeur )

I will speak about multidimensional shifts of finite type and their measures of maximal entropy. In particular, I will present results about computability of topological entropy for SFTs and measure-theoretic entropy. I'll focus on various mixing hypotheses, both topological and measure-theoretic, which imply different rates of computability for these objects, and give applications to various systems, including the hard square model, k-coloring, and iceberg model. I will speak about multidimensional shifts of finite type and their measures of maximal entropy. In particular, I will present results about computability of topological entropy for SFTs and measure-theoretic entropy. I'll focus on various mixing hypotheses, both topological and measure-theoretic, which imply different rates of computability for these objects, and give applications to various systems, including the hard square model, k-coloring, ...

## Post-edited  Entropy and mixing for multidimensional shifts of finite type - Lecture 2 Pavlov, Ronnie (Auteur de la Conférence) | CIRM (Editeur )

I will speak about multidimensional shifts of finite type and their measures of maximal entropy. In particular, I will present results about computability of topological entropy for SFTs and measure-theoretic entropy. I'll focus on various mixing hypotheses, both topological and measure-theoretic, which imply different rates of computability for these objects, and give applications to various systems, including the hard square model, k-coloring, and iceberg model. I will speak about multidimensional shifts of finite type and their measures of maximal entropy. In particular, I will present results about computability of topological entropy for SFTs and measure-theoretic entropy. I'll focus on various mixing hypotheses, both topological and measure-theoretic, which imply different rates of computability for these objects, and give applications to various systems, including the hard square model, k-coloring, ...

## Post-edited  Entropy and mixing for multidimensional shifts of finite type - Lecture 3 Pavlov, Ronnie (Auteur de la Conférence) | CIRM (Editeur )

I will speak about multidimensional shifts of finite type and their measures of maximal entropy. In particular, I will present results about computability of topological entropy for SFTs and measure-theoretic entropy. I'll focus on various mixing hypotheses, both topological and measure-theoretic, which imply different rates of computability for these objects, and give applications to various systems, including the hard square model, k-coloring, and iceberg model. I will speak about multidimensional shifts of finite type and their measures of maximal entropy. In particular, I will present results about computability of topological entropy for SFTs and measure-theoretic entropy. I'll focus on various mixing hypotheses, both topological and measure-theoretic, which imply different rates of computability for these objects, and give applications to various systems, including the hard square model, k-coloring, ...

## Post-edited  Dense subgroups in simple groups - Lecture 2 Benoist, Yves (Auteur de la Conférence) | CIRM (Editeur )

In this series of lectures, we will focus on simple Lie groups, their dense subgroups and the convolution powers of their measures. In particular, we will dicuss the following two questions.
Let G be a Lie group. Is every Borel measurable subgroup of G with maximal Hausdorff dimension equal to the group G?
Is the convolution of sufficiently many compactly supported continuous functions on G always continuously differentiable?
Even though the answer to these questions is no when G is abelian, the answer is yes when G is simple. This is a joint work with N. de Saxce. First, I will explain the history of these two questions and their interaction. Then, I will relate these questions to spectral gap properties. Finally, I will discuss these spectral gap properties.
In this series of lectures, we will focus on simple Lie groups, their dense subgroups and the convolution powers of their measures. In particular, we will dicuss the following two questions.
Let G be a Lie group. Is every Borel measurable subgroup of G with maximal Hausdorff dimension equal to the group G?
Is the convolution of sufficiently many compactly supported continuous functions on G always continuously differentiable?
Even though the ...

## Post-edited  Numerical methods for mean field games - Lecture 2: Monotone finite difference schemes Achdou, Yves (Auteur de la Conférence) | CIRM (Editeur )

Recently, an important research activity on mean field games (MFGs for short) has been initiated by the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number $n$ of agents tends to infinity. The field is now rapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculus of variations, numerical analysis and computing, and the potential applications to economics and social sciences are numerous.
In the limit when $n \to +\infty$, a given agent feels the presence of the others through the statistical distribution of the states. Assuming that the perturbations of a single agent's strategy does not influence the statistical states distribution, the latter acts as a parameter in the control problem to be solved by each agent. When the dynamics of the agents are independent stochastic processes, MFGs naturally lead to a coupled system of two partial differential equations (PDEs for short), a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation.
The latter system of PDEs has closed form solutions in very few cases only. Therefore, numerical simulation are crucial in order to address applications. The present mini-course will be devoted to numerical methods that can be used to approximate the systems of PDEs.
The numerical schemes that will be presented rely basically on monotone approximations of the Hamiltonian and on a suitable weak formulation of the Fokker-Planck equation.
These schemes have several important features:

- The discrete problem has the same structure as the continous one, so existence, energy estimates, and possibly uniqueness can be obtained with the same kind of arguments

- Monotonicity guarantees the stability of the scheme: it is robust in the deterministic limit

- convergence to classical or weak solutions can be proved

Finally, there are particular cases named variational MFGS in which the system of PDEs can be seen as the optimality conditions of some optimal control problem driven by a PDE. In such cases, augmented Lagrangian methods can be used for solving the discrete nonlinear system. The mini-course will be orgamized as follows

1. Introduction to the system of PDEs and its interpretation. Uniqueness of classical solutions.

2. Monotone finite difference schemes

3. Examples of applications

4. Variational MFG and related algorithms for solving the discrete system of nonlinear equations
Recently, an important research activity on mean field games (MFGs for short) has been initiated by the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number $n$ of agents tends to infinity. The field is now rapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculus of variations, numerical analysis and ...

## Multi angle  Compter et optimiser avec les graphes unimodulaires - Cours 1 Bordenave, Charles (Auteur de la Conférence) | CIRM (Editeur )

L'objectif de ce mini-cours est de présenter de la façon la plus élémentaire possible la convergence faible locale des graphes introduite par Benjamini et Schramm en 2001 et développée par Aldous et Steele (2004), Aldous et Lyons (2007). Nous montrerons comment cette notion peut être utilisée dans des dénombrements asymptotiques et dans des problèmes d'optimisation combinatoire.

## Multi angle  Compter et optimiser avec les graphes unimodulaires - Cours 2 Bordenave, Charles (Auteur de la Conférence) | CIRM (Editeur )

L'objectif de ce mini-cours est de présenter de la façon la plus élémentaire possible la convergence faible locale des graphes introduite par Benjamini et Schramm en 2001 et développée par Aldous et Steele (2004), Aldous et Lyons (2007). Nous montrerons comment cette notion peut être utilisée dans des dénombrements asymptotiques et dans des problèmes d'optimisation combinatoire.

## Multi angle  Le diamant aztèque - Cours 1 Corteel, Sylvie (Auteur de la Conférence) | CIRM (Editeur )

Le but du mini-cours sera de faire un cours introductif à différentes méthodes énumeratives à travers l'exemple des pavages par dominos du diamant aztèque. On essaiera de voir les fonctions (super)-symétriques, les moments de polynômes bi-orthogonaux, les évaluations de determinants, les algorithmes de génération...

## Multi angle  Le diamant aztèque - Cours 2 Corteel, Sylvie (Auteur de la Conférence) | CIRM (Editeur )

Le but du mini-cours sera de faire un cours introductif à différentes méthodes énumeratives à travers l'exemple des pavages par dominos du diamant aztèque. On essaiera de voir les fonctions (super)-symétriques, les moments de polynômes bi-orthogonaux, les évaluations de determinants, les algorithmes de génération...

## Multi angle  Reconstruction methods for ill-posed inverse problems - Part 1 Siltanen, Samuli (Auteur de la Conférence) | CIRM (Editeur )

inverse problem - reconstruction - regularization - tomography - computation

## Multi angle  Reconstruction methods for ill-posed inverse problems - Part 2 Siltanen, Samuli (Auteur de la Conférence) | CIRM (Editeur )

inverse problem - reconstruction - regularization - tomography - computation

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