Research schools  | enregistrements trouvés : 112

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, 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, and iceberg model.

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.

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

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.

We consider the convergence of the iterative projected gradient (IPG) algorithm for arbitrary (typically nonconvex) sets and when both the gradient and projection oracles are only computed approximately. We consider different notions of approximation of which we show that the Progressive Fixed Precision (PFP) and (1+epsilon) optimal oracles can achieve the same accuracy as for the exact IPG algorithm. We also show that the former scheme is also able to maintain the (linear) rate of convergence of the exact algorithm, under the same embedding assumption, while the latter requires a stronger embedding condition, moderate compression ratios and typically exhibits slower convergence. We apply our results to accelerate solving a class of data driven compressed sensing problems, where we replace iterative exhaustive searches over large datasets by fast approximate nearest neighbour search strategies based on the cover tree data structure. Finally, if there is time we will give examples of this theory applied in practice for rapid enhanced solutions to an emerging MRI protocol called magnetic resonance fingerprinting for quantitative MRI.

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.

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.

The question of using the available measurements to retrieve mathematical models characteristics (parameters, boundary conditions, initial conditions) is a key aspect of the modeling objective in biology or medicine. In a stochastic/statistical framework this question is seen as an estimation problems. From a deterministic point of view, we classical talk about inverse problems as we recover classical model inputs from outputs. When considering evolution problems,this question falls in the realm of data assimilation that can be seen from a deterministic of statistical point of view. Our objective in this course is to introduce the mathematical principles and numerical aspects behind data assimilation strategies with an emphasis on the deterministic formalism allowing to understand why data assimilation is a specific inverse problem. Our presentation will include considerations on finite dimensional problems but also on infinite dimensional problems such as the ones arising from PDE models. And we will illustrate the course with numerous examples coming from cardiovascular applications and biology.

This minicourse aims at providing tentative explanations of some specific phenomena observed in the motion of crowds, or more generally collections of living entities. The first lecture shall focus on the so-called Stop and Go Waves, which sometimes spontaneously emerge and persist in crowds in motion. We shall present a general class of dynamical systems which are likely to exhibit this type of instabilities, and emphasize the critical role of two basic ingredients: the asymmetry of interactions, and any sort of delay in the transmission of information through the network of entities. The second lecture will address the Capacity Drop Phenomenon (decrease of the flux though a bottleneck when the upstream density becomes too high), and the more paradoxical Faster is Slower Effect (in some regimes, attempts to go quicker may slow down the overall process). We shall in particular detail how an accurate description of the relative position of entities (at the microscopic level) is crucial to recover and understand those effects.

Irreversible electroporation (IRE) is the sole physical ablative technology inducing tumorous cell death by process unrelated to thermal effect. This characteristic makes the technique suitable for the treatment of subtypes of liver tumors especially hepatocellular carcinoma (HCC) located next to critical structures leading to contraindications to thermal ablation like radiofrequency, microwave or cryotherapy. However, while IRE appears safe in such assumed challenging cases for thermal techniques, several issues remain to be addressed to make its use easier and more effective in clinical practice. First of all, tissue changes induced by IRE must be assessed keeping in mind that conversely to thermal techniques its efficacy is not limited to observable coagulative necrotic component of treatment zone. In addition, IRE which is multibipolar ablative technology requires meticulous demanding electrodes positioning to ensure proper magnitude of electric fields between each dipole. Finally, numerical simulations of IRE are mandatory to ease the setting of electrical pulses parameters to improve predictability of treatment in each individual case. In this setting of continue efforts to improve practicability of IRE the technique is routinely used in our institution since several years for the treatment of patients bearing early and locally advanced HCC not amenable to resection or thermal ablation. All along our experience with IRE, imaging appeared as a key point for addressing the specific issues listed above. For the 58 first patients 92% of complete ablation were achieved while the one-year local tumor progression free survival was 70% (95% CI: 56%, 81%). Indeed, despite the need of improvements IRE appears right now as a unique opportunity to achieve complete sustained local tumor control for patient bearing early or locally advanced HCC not amenable to other curative treatments.