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Documents : Post-edited  Conférences Vidéo Chapitrées | enregistrements trouvés : 200

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In this talk we discuss the convergence to equilibrium in conservative-dissipative ODE-systems, kinetic relaxation models (of BGK-type), and Fokker-Planck equation. This will include symmetric, non-symmetric and hypocoercive evolution equations. A main focus will be on deriving sharp decay rates.
We shall start with hypocoercivity in ODE systems, with the ”hypocoercivity index” characterizing its structural complexity.
BGK equations are kinetic transport equations with a relaxation operator that drives the phase space distribution towards the spatially local equilibrium, a Gaussian with the same macroscopic parameters. Due to the absence of dissipation w.r.t. the spatial direction, convergence to the global equilibrium is only possible thanks to the transport term that mixes various positions. Hence, such models are hypocoercive.
We shall prove exponential convergence towards the equilibrium with explicit rates for several linear, space periodic BGK-models in dimension 1 and 2. Their BGK-operators differ by the number of conserved macroscopic quantities (like mass, momentum, energy), and hence their hypocoercivity index. Our discussion includes also discrete velocity models, and the local exponential stability of a nonlinear BGK-model.
The third part of the talk is concerned with the entropy method for (non)symmetric Fokker-Planck equations, which is a powerful tool to analyze the rate of convergence to the equilibrium (in relative entropy and hence in L1). The essence of the method is to first derive a differential inequality between the first and second time derivative of the relative entropy, and then between the entropy dissipation and the entropy. For hypocoercive Fokker-Planck equations, i.e. degenerate parabolic equations (with drift terms that are linear in the spatial variable) we modify the classical entropy method by introducing an auxiliary functional (of entropy dissipation type) to prove exponential decay of the solution towards the steady state in relative entropy. The obtained rate is indeed sharp (both for the logarithmic and quadratic entropy). Finally, we extend the method to the kinetic Fokker-Planck equation (with nonquadratic potential).
In this talk we discuss the convergence to equilibrium in conservative-dissipative ODE-systems, kinetic relaxation models (of BGK-type), and Fokker-Planck equation. This will include symmetric, non-symmetric and hypocoercive evolution equations. A main focus will be on deriving sharp decay rates.
We shall start with hypocoercivity in ODE systems, with the ”hypocoercivity index” characterizing its structural complexity.
BGK equations are kinetic ...

35Q84 ; 35H10 ; 35B20 ; 35K10 ; 35B40 ; 47D07 ; 35Pxx ; 47D06 ; 82C31

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Post-edited  Mahler's method in several variables
Adamczewski, Boris (Auteur de la Conférence) | CIRM (Editeur )

Any algebraic (resp. linear) relation over the field of rational functions with algebraic coefficients between given analytic functions leads by specialization to algebraic (resp. linear) relations over the field of algebraic numbers between the values of these functions. Number theorists have long been interested in proving results going in the other direction. Though the converse result is known to be false in general, Mahler’s method provides one of the few known instances where it essentially holds true. After the works of Nishioka, and more recently of Philippon, Faverjon and the speaker, the theory of Mahler functions in one variable is now rather well understood. In contrast, and despite the contributions of Mahler, Loxton and van der Poorten, Kubota, Masser, and Nishioka among others, the theory of Mahler functions in several variables remains much less developed. In this talk, I will discuss recent progresses concerning the case of regular singular systems, as well as possible applications of this theory. This is a joint work with Colin Faverjon.
Any algebraic (resp. linear) relation over the field of rational functions with algebraic coefficients between given analytic functions leads by specialization to algebraic (resp. linear) relations over the field of algebraic numbers between the values of these functions. Number theorists have long been interested in proving results going in the other direction. Though the converse result is known to be false in general, Mahler’s method provides ...

11J81 ; 11J85 ; 11B85

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In many situations where stochastic modeling is used, one desires to choose the coefficients of a stochastic differential equation which represents the reality as simply as possible. For example one desires to approximate a diffusion model
with high complexity coefficients by a model within a class of simple diffusion models. To achieve this goal, we introduce a new Wasserstein type distance on the set of laws of solutions to d-dimensional stochastic differential equations.
This new distance $\widetilde{W}^{2}$ is defined similarly to the classical Wasserstein distance $\widetilde{W}^{2}$ but the set of couplings is restricted to the set of laws of solutions of 2$d$-dimensional stochastic differential equations. We prove that this new distance $\widetilde{W}^{2}$ metrizes the weak topology. Furthermore this distance $\widetilde{W}^{2}$ is characterized in terms of a stochastic control problem. In the case d = 1 we can construct an explicit solution. The multi-dimensional case, is more tricky and classical results do not apply to solve the HJB equation because of the degeneracy of the differential operator. Nevertheless, we prove that this HJB equation admits a regular solution.
In many situations where stochastic modeling is used, one desires to choose the coefficients of a stochastic differential equation which represents the reality as simply as possible. For example one desires to approximate a diffusion model
with high complexity coefficients by a model within a class of simple diffusion models. To achieve this goal, we introduce a new Wasserstein type distance on the set of laws of solutions to d-dimensional ...

91B70 ; 60H30 ; 60H15 ; 60J60 ; 93E20

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Graphons and graphexes are limits of graphs which allow us to model and estimate properties of large-scale networks. In this pair of talks, we review the theory of dense graph limits, and give two alterative theories for limits of sparse graphs - one leading to unbounded graphons over probability spaces, and the other leading to bounded graphons (and graphexes) over sigma-finite measure spaces. Talk I, given by Jennifer, will review the general theory, highlight the unbounded graphons, and show how they can be used to consistently estimate properties of large sparse networks. This talk will also give an application of these sparse graphons to collaborative filtering on sparse bipartite networks. Talk II, given by Christian, will recast limits of dense graphs in terms of exchangeability and the Aldous Hoover Theorem, and generalize this to obtain sparse graphons and graphexes as limits of subgraph samples from sparse graph sequences. This will provide a dual view of sparse graph limits as processes and random measures, an approach which allows a generalization of many of the well-known results and techniques for dense graph sequences.
Graphons and graphexes are limits of graphs which allow us to model and estimate properties of large-scale networks. In this pair of talks, we review the theory of dense graph limits, and give two alterative theories for limits of sparse graphs - one leading to unbounded graphons over probability spaces, and the other leading to bounded graphons (and graphexes) over sigma-finite measure spaces. Talk I, given by Jennifer, will review the general ...

05C80 ; 05C60 ; 60F10 ; 82B20

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Graphons and graphexes are limits of graphs which allow us to model and estimate properties of large-scale networks. In this pair of talks, we review the theory of dense graph limits, and give two alterative theories for limits of sparse graphs - one leading to unbounded graphons over probability spaces, and the other leading to bounded graphons (and graphexes) over sigma-finite measure spaces. Talk I, given by Jennifer, will review the general theory, highlight the unbounded graphons, and show how they can be used to consistently estimate properties of large sparse networks. This talk will also give an application of these sparse graphons to collaborative filtering on sparse bipartite networks. Talk II, given by Christian, will recast limits of dense graphs in terms of exchangeability and the Aldous Hoover Theorem, and generalize this to obtain sparse graphons and graphexes as limits of subgraph samples from sparse graph sequences. This will provide a dual view of sparse graph limits as processes and random measures, an approach which allows a generalization of many of the well-known results and techniques for dense graph sequences.
Graphons and graphexes are limits of graphs which allow us to model and estimate properties of large-scale networks. In this pair of talks, we review the theory of dense graph limits, and give two alterative theories for limits of sparse graphs - one leading to unbounded graphons over probability spaces, and the other leading to bounded graphons (and graphexes) over sigma-finite measure spaces. Talk I, given by Jennifer, will review the general ...

05C80 ; 05C60 ; 60F10 ; 82B20

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Post-edited  Of commutators and Jacobians
Hytönen, Tuomas P. (Auteur de la Conférence) | CIRM (Editeur )

The boundedness (on $L^p$ spaces) of commutators $[b,T] = bT-Tb$ of pointwise multiplication $b$ and singular integral operators $T$ has been well studied for a long time. Curiously, the necessary conditions for this boundedness to happen are generally less understood than the sufficient conditions, for instance what comes to the assumptions on the operator $T$. I will discuss some new results in this direction, and show how this circle of ideas relates to the mapping properties of the Jacobian (the determinant of the derivative matrix) on first order Sobolev spaces. This is work in progress at the time of submitting the abstract, so I will hopefully be able to present some fairly fresh material.
The boundedness (on $L^p$ spaces) of commutators $[b,T] = bT-Tb$ of pointwise multiplication $b$ and singular integral operators $T$ has been well studied for a long time. Curiously, the necessary conditions for this boundedness to happen are generally less understood than the sufficient conditions, for instance what comes to the assumptions on the operator $T$. I will discuss some new results in this direction, and show how this circle of ideas ...

42B20 ; 42B35

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Post-edited  Morsifications and mutations
Fomin, Sergey (Auteur de la Conférence) | CIRM (Editeur )

I will discuss a connection between the topology of isolated singularities of plane curves and the mutation equivalence of the quivers associated with their morsifications. Joint work with Pavlo Pylyavskyy, Eugenii Shustin, and Dylan Thurston.

13F60 ; 20F36 ; 57M25 ; 58K65

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Post-edited  Interview at Cirm: Shigeki Akiyama
Akiyama, Shigeki (Personne interviewée) | CIRM (Editeur )

Shigeki Akiyama is Professor at the Institute of Mathematics of the University of Tsukuba, Japan. A regular organizer of the annual workshop on quasi-periodic tilings at RIMS, he has also spent time as organizer or invited professor on several occasions in France (Paris, Marseille, Strasbourg) but also in Debrecen and at the Chinese University of Hong-Kong.
CIRM - Chaire Jean-Morlet 2017 - Aix-Marseille Université

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Graph searching, a mechanism to traverse a graph visiting one vertex at a time in a specific manner, is a powerful tool used to extract structure from various families of graphs. In this talk, we focus on two graph searches: Lexicographic Breadth First Search (LBFS), and Lexicographic Depth First Search (LDFS).
Many classes of graphs have a vertex ordering characterisation, and we review how graph searching is used to produce efficiently such vertex orderings.
These orderings expose structure that we exploit to develop efficient linear and near-linear time algorithms for some NP-hard problems (independent set, colouring, Hamiltonicity for instance) on some special classes of graphs such as cocomparability graphs.
In particular, we will prove fixed point type theorems for LexBFS, and then focus on a LexDFS-based framework to lift algorithms from interval graphs to cocomparability graphs. Then I will present the relationships between graph searches, graph geometric convexities and antimatroids. These relationships are for to be completely understood and I will pose some hard conjectures and some interesting problems to consider.
To finish I will present some recent results about Robinsonian matrices by M. Laurent and M. Seminaroti and their relationships with graph searches. This yields a new area of research to investigate.
Graph searching, a mechanism to traverse a graph visiting one vertex at a time in a specific manner, is a powerful tool used to extract structure from various families of graphs. In this talk, we focus on two graph searches: Lexicographic Breadth First Search (LBFS), and Lexicographic Depth First Search (LDFS).
Many classes of graphs have a vertex ordering characterisation, and we review how graph searching is used to produce efficiently such ...

05C85 ; 68R10

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Post-edited  Coarse dimension reduction
Naor, Assaf (Auteur de la Conférence) | CIRM (Editeur )

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Given an automorphism of the free group, we consider the mapping torus defined with respect to the automorphism. If the automorphism is atoroidal, then the resulting free-by-cyclic group is hyperbolic by work of Brinkmann. In addition, if the automorphism is fully irreducible, then work of Kapovich-Kleiner proves the boundary of the group is homeomorphic to the Menger curve. However, their proof is very general and gives no tools to further study the boundary and large-scale geometry of these groups. In this talk, I will explain how to construct explicit embeddings of non-planar graphs into the boundary of these groups whenever the group is hyperbolic. Along the way, I will illustrate how our methods distinguish free-by-cyclic groups which are the fundamental group of a 3-manifold. This is joint work with Yael Algom-Kfir and Arnaud Hilion.
Given an automorphism of the free group, we consider the mapping torus defined with respect to the automorphism. If the automorphism is atoroidal, then the resulting free-by-cyclic group is hyperbolic by work of Brinkmann. In addition, if the automorphism is fully irreducible, then work of Kapovich-Kleiner proves the boundary of the group is homeomorphic to the Menger curve. However, their proof is very general and gives no tools to further ...

20F65 ; 20F67 ; 20E36

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Let $G$ be a torsion-free hyperbolic group, let $S$ be a finite generating set of $G$, and let $f$ be an automorphism of $G$. We want to understand the possible growth types for the word length of $f^n(g)$, where $g$ is an element of $G$. Growth was completely described by Thurston when $G$ is the fundamental group of a hyperbolic surface, and can be understood from Bestvina-Handel's work on train-tracks when $G$ is a free group. We address the general case of a torsion-free hyperbolic group $G$; we show that every element in $G$ has a well-defined exponential growth rate under iteration of $f$, and that only finitely many exponential growth rates arise as $g$ varies in $G$. In addition, we show the following dichotomy: every element of $G$ grows either exponentially fast or polynomially fast under iteration of $f$.
This is a joint work with Rémi Coulon, Arnaud Hilion and Gilbert Levitt.
Let $G$ be a torsion-free hyperbolic group, let $S$ be a finite generating set of $G$, and let $f$ be an automorphism of $G$. We want to understand the possible growth types for the word length of $f^n(g)$, where $g$ is an element of $G$. Growth was completely described by Thurston when $G$ is the fundamental group of a hyperbolic surface, and can be understood from Bestvina-Handel's work on train-tracks when $G$ is a free group. We address the ...

57M07 ; 20E06 ; 20F34 ; 20F65 ; 20E36 ; 20F67

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In this talk, we will develop the theory of generalized bridge trisections for smoothly embedded closed surfaces in smooth, closed four-manifolds. The main result is that any such surface can be isotoped to lie in bridge trisected position with respect to a given trisection of the ambient four-manifold. In the setting of knotted surfaces in the four-sphere, this gives a diagrammatic calculus that offers a promising new approach to four-dimensional knot theory. However, the theory extends to other ambient four-manifolds, and we will pay particular attention to the setting of complex curves in simple complex surfaces, where the theory produces surprisingly satisfying pictures and leads to interesting results about trisections of complex surfaces.
This talk is based on various joint works with Dave Gay, Peter Lambert-Cole, and Alex Zupan.
In this talk, we will develop the theory of generalized bridge trisections for smoothly embedded closed surfaces in smooth, closed four-manifolds. The main result is that any such surface can be isotoped to lie in bridge trisected position with respect to a given trisection of the ambient four-manifold. In the setting of knotted surfaces in the four-sphere, this gives a diagrammatic calculus that offers a promising new approach to four-...

57Q45 ; 57M25 ; 57M50

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In this course I will first introduce cluster algebras associated with a triangulated surface. I will then focus on representation of quivers, and show the strong link between cluster combinatorics and representation theory. The aim will be to explain additive categorification of cluster algebras in this context. All the notions will be illustrated by examples.

13F60 ; 16E35 ; 16G20 ; 18E30

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The Reidemeister torsion may be viewed as a volume form on the character variety of a 3-manifold with boundary. I will explain a conjectural differential equation that this form should satisfy, motivated by the study of the asymptotical behaviour of quantum invariants.

53D50 ; 57M25 ; 57M27 ; 57R56

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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 ...

53-04 ; 53Axx ; 58C25 ; 68N01 ; 68N15 ; 68U05

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In the 80's Beauville generalized several foundational results of Nikulin on automorphism groups of K3 surfaces to hyperkähler manifolds. Since then the study of automorphism groups of hyperkähler manifolds and in particular of hyperkähler fourfolds got very much attention. I will present some classification results for automorphisms on hyperkähler fourfolds that are deformation equivalent to the Hilbert scheme of two points on a K3 surface and describe some explicit examples. I will give particular attention to double EPW sextics, that admit in a natural way a non-symplectic involution. Time permitting I will show how the rich geometry of double EPW sextics has an important connection to a classical question of U. Morin (1930).
In the 80's Beauville generalized several foundational results of Nikulin on automorphism groups of K3 surfaces to hyperkähler manifolds. Since then the study of automorphism groups of hyperkähler manifolds and in particular of hyperkähler fourfolds got very much attention. I will present some classification results for automorphisms on hyperkähler fourfolds that are deformation equivalent to the Hilbert scheme of two points on a K3 surface and ...

14J50 ; 14J28 ; 14J35 ; 14J70 ; 14M15 ; 14N20

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Bounded remainder sets for a dynamical system are sets for which the Birkhoff averages of return times differ from the expected values by at most a constant amount. These sets are rare and important objects which have been studied for over 100 years. In the last few years there have been a number of results which culminated in explicit constructions of bounded remainder sets for toral rotations in any dimension, of all possible allowable volumes. In this talk we are going to explain these results, and then explain how to generalize them to give explicit constructions of bounded remainder sets for rotations in $p$-adic solenoids. Our method of proof will make use of a natural dynamical encoding of patterns in non-Archimedean cut and project sets.
Bounded remainder sets for a dynamical system are sets for which the Birkhoff averages of return times differ from the expected values by at most a constant amount. These sets are rare and important objects which have been studied for over 100 years. In the last few years there have been a number of results which culminated in explicit constructions of bounded remainder sets for toral rotations in any dimension, of all possible allowable ...

11K06 ; 11K38 ; 11J71

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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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