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Research schools  | enregistrements trouvés : 109

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These lectures introduce the dynamical systems approach to tilings of Euclidean space, especially quasicrystalline tilings that have been constructed using a ‘supertile method’. Because tiling dynamics parallels one-dimensional symbolic dynamics, we discuss this case as well, highlighting the differences and similarities in the methods of study and the results that can be obtained.
In the first lecture we motivate the field with the discovery of quasicrystals, which led to D. Schectman’s winning the 2011 Nobel Prize in Chemistry. Then we set up the basics of tiling dynamics, describing tiling spaces, a tiling metric, and the shift or translation actions. Shift-invariant and ergodic measures are discussed, along with fundamental topological and dynamical properties.
The second lecture brings in the supertile construction methods, including symbolic substitutions, self-similar tilings, $S$-adic systems, and fusion rules. Numerous examples are given, most of which are not the “standard” examples, and we identify many commonalities and differences between these interrelated methods of construction. Then we compare and contrast dynamical results for supertile systems, highlighting those key insights that can be adapted to all cases.
In the third lecture we investigate one of the many current tiling research areas: spectral theory. Schectman made his Nobel-prize-winning discovery using diffraction analysis, and studying the mathematical version has been quite fruitful. Spectral theory of tiling dynamical systems is also of broad interest. We describe how these types of spectral analysis are carried out, give examples, and discuss what is known and unknown about the relationship between dynamical and diffraction analysis. Special attention is paid to the “point spectrum”, which is related to eigenfunctions and also to the bright spots that appear on diffraction images.
These lectures introduce the dynamical systems approach to tilings of Euclidean space, especially quasicrystalline tilings that have been constructed using a ‘supertile method’. Because tiling dynamics parallels one-dimensional symbolic dynamics, we discuss this case as well, highlighting the differences and similarities in the methods of study and the results that can be obtained.
In the first lecture we motivate the field with the discovery of ...

37B50 ; 37B10 ; 52C23

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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 ...

49K20 ; 49N70 ; 35K40 ; 35K55 ; 35Q84 ; 65K10 ; 65M06 ; 65M12 ; 91A23 ; 91A15

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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 ...

22E30 ; 28A78 ; 43A65

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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, ...

37B50 ; 37B10 ; 37B40

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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, ...

37B50 ; 37B10 ; 37B40

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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, ...

37B50 ; 37B10 ; 37B40

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- 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.

05E10 ; 05E15 ; 20C30 ; 05A15 ; 05C10

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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.

11B85 ; 68Q45 ; 68R15

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Spherical Hecke algebra, Satake transform, and an introduction to local Langlands correspondence.
CIRM - Chaire Jean-Morlet 2016 - Aix-Marseille Université

20C08 ; 22E50 ; 11S37

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Post-edited  Coloring graphs on surfaces
Esperet, Louis (Auteur de la Conférence) | CIRM (Editeur )

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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.

81T30 ; 83E30

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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, ...

60G18 ; 60J25 ; 60J85

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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

37Cxx ; 37Jxx ; 53D25 ; 53D40 ; 53D42

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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 ...

35R30 ; 35L05 ; 35L20

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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. ...

60D05 ; 60F05 ; 52A22 ; 60G55

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The general aim of these lectures is to present some interplay between combinatorial game theory (CGT) and combinatorics on (multidimensional) words.
In the first introductory lecture, we present some basic concepts from combinatorial game theory (positions of a game, Nim-sum, Sprague-Grundy function, Wythoff’s game, ...). We also review some concepts from combinatorics on words. We thus introduce the well-known k-automatic sequences and review some of their characterizations (in terms of morphisms, finiteness of their k-kernel,...). These sequences take values in a finite set but the Sprague-Grundy function of a game, such as Nim of Wythoff, is usually unbounded. This provides a motivation to introduce k-regular sequences (in the sense of Allouche and Shallit) whose k-kernel is not finite, but finitely generated.
In the second lecture, games played on several piles of token naturally give rise to a multi-dimensional setting. Thus, we reconsider k-automatic and k-regular sequences in this extended framework. In particular, determining the structure of the bidimensional array encoding the (loosing) P-positions of the Wythoff’s game is a long-standing and challenging problem in CGT. Wythoff’s game is linked to non-standard numeration system: P-positions can be determined by writing those in the Fibonacci system. Next, we present the concept of shape-symmetric morphism: instead of iterating a morphism where images of letters are (hyper-)cubes of a fixed length k, one can generalize the procedure to images of various parallelepipedic shape. The shape-symmetry condition introduced twenty years ago by Maes permits to have well-defined fixed point.
In the last lecture, we move to generalized numeration systems: abstract numeration systems (built on a regular language) and link them to morphic (multidimensional) words. In particular, pictures obtained by shape-symmetric morphisms coincide with automatic sequences associated with an abstract numeration system. We conclude these lectures with some work in progress about games with a finite rule-set. This permits us to discuss a bit Presburger definable sets.
The general aim of these lectures is to present some interplay between combinatorial game theory (CGT) and combinatorics on (multidimensional) words.
In the first introductory lecture, we present some basic concepts from combinatorial game theory (positions of a game, Nim-sum, Sprague-Grundy function, Wythoff’s game, ...). We also review some concepts from combinatorics on words. We thus introduce the well-known k-automatic sequences and review ...

91A46 ; 68R15 ; 68Q45

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Multi angle  Lecture on Delone sets and tilings
Solomyak, Boris (Auteur de la Conférence) | CIRM (Editeur )

In this lecture we focus on selected topics around the themes: Delone sets as models for quasicrystals, inflation symmetries and expansion constants, substitution Delone sets and tilings, and associated dynamical systems.

52C23 ; 37B50

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Based on work done by Morse and Hedlund (1940) it was observed by Arnoux and Rauzy (1991) that the classical continued fraction algorithm provides a surprising link between arithmetic and diophantine properties of an irrational number $\alpha$, the rotation by $\alpha$ on the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, and combinatorial properties of the well known Sturmian sequences, a class of sequences on two letters with low subword complexity.
It has been conjectured since the early 1990ies that this correspondence carries over to generalized continued fraction algorithms, rotations on higher dimensional tori, and so-called $S$-adic sequences generated by substitutions. The idea of working towards this generalization is known as Rauzy’s program. Although, starting with Rauzy (1982) a number of examples for such a generalization was devised, Cassaigne, Ferenczi, and Zamboni (2000) came up with a counterexample that showed the limitations of such a generalization.
Nevertheless, recently Berthé, Steiner, and Thuswaldner (2016) made some further progress on Rauzy’s program and were able to set up a generalization of the above correspondences. They proved that the above conjecture is true under certain natural conditions. A prominent role in this generalization is played by tilings induced by generalizations of the classical Rauzy fractal introduced by Rauzy (1982).
Another idea which is related to the above results goes back to Artin (1924), who observed that the classical continued fraction algorithm and its natural extension can be viewed as a Poincaré section of the geodesic flow on the space $SL_2(\mathbb{Z}) \ SL_2(\mathbb{R})$. Arnoux and Fisher (2001) revisited Artin’s idea and showed that the above mentioned correspondence between continued fractions, rotations, and Sturmian sequences can be interpreted in a very nice way in terms of an extension of this geodesic flow which they called the scenery flow. Currently, Arnoux et al. are setting up elements of a generalization of this connection as well.
It is the aim of my series of lectures to review the above results.
Based on work done by Morse and Hedlund (1940) it was observed by Arnoux and Rauzy (1991) that the classical continued fraction algorithm provides a surprising link between arithmetic and diophantine properties of an irrational number $\alpha$, the rotation by $\alpha$ on the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, and combinatorial properties of the well known Sturmian sequences, a class of sequences on two letters with low subword ...

11B83 ; 11K50 ; 37B10 ; 52C23 ; 53D25

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One of the most fundamental problem in tiling theory is to decide, given a surface, a set of tiles and a tiling rule, whether there exist a way to tile the surface using the set of tiles and following the rules. As proven by Berger in the 60’s, this problem is undecidable in general.
When formulated in terms of tilings of the discrete plane by unit tiles with colored constraints, this is called the Domino Problem and was introduced by Wang in an effort to solve satisfaction problems for ??? formulas by translating the problem into a geometric problem.
In this course, we will give a brief description of the problem and to the meaning of the word “undecidable”, and then give two different proofs of the result.
One of the most fundamental problem in tiling theory is to decide, given a surface, a set of tiles and a tiling rule, whether there exist a way to tile the surface using the set of tiles and following the rules. As proven by Berger in the 60’s, this problem is undecidable in general.
When formulated in terms of tilings of the discrete plane by unit tiles with colored constraints, this is called the Domino Problem and was introduced by Wang in an ...

03D35 ; 05B45

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