Colloquium du CEREMADE

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Intervenant : FRANK Rupert (Ludwig Maximilians Universität München)
Titre :
Sharp functional inequalities and their stability

Le : 06/05/2025 de : 15:30 à : 16:30
The Sobolev inequality is a paradigmatic example of a functional inequality with many applications in the Calculus of Variations, Geometric Analysis and PDEs. In some of these applications the optimal value of the constant is of importance, as is a characterization of the set of optimizers. The stability question is whether functions whose Sobolev quotient is almost minimal are close to minimizers of the inequality and, if so, in which sense. We give a gentle introduction to this question and review some recent results on the Sobolev inequality and other functional inequalities of a similar nature.
Salle : A709

Colloquium du CEREMADE

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Intervenant : REYNAUD-BOURET Patricia (Université Côte d'Azur)
Titre :
Kalikow decomposition for the study of neuronal networks: simulation and learning

Le : 01/04/2025 de : 15:30 à : 16:30
Kalikow decomposition is a decomposition of stochastic processes (usually finite state discrete time processes but also more recently point processes) that consists in picking at random a finite neighborhood in the past and then make a transition in a Markov manner. This kind of approach has been used for many years to prove existence of some processes, especially their stationary distribution. In particular, it allows to prove the existence of processes that model infinite neuronal networks, such as Hawkes like processes or Galvès-Löcherbach processes. But beyond mere existence, this decomposition is a wonderful tool to simulate such network, as an open physical system, that from a computational point of view could be competitive with the most performant brain simulations. This decomposition is also a source of inspiration to understand how local rules at each neuron can make the whole network learn.
Salle : A709

Colloquium du CEREMADE

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Intervenant : WANG Yilin (IHES)
Titre :
The Brownian loop measure on Riemann surfaces and applications to length spectra

Le : 04/03/2025 de : 15:30 à : 16:30
The goal of this talk is to showcase how we can use stochastic processes to study the geometry of surfaces. More precisely, we use the Brownian loop measure to express the lengths of closed geodesics on a hyperbolic surface and zeta-regularized determinant of the Laplace-Beltrami operator. This gives a tool to study the length spectra of a hyperbolic surface and we obtain a new identity between the length spectrum of a compact surface and that of the same surface with an arbitrary number of additional cusps. This is based on a joint work with Yuhao Xue (IHES).
Salle : A709

Colloquium du CEREMADE

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Intervenant : DALIBARD Anne-Laure (Sorbonne Université)
Titre :
Stationary boundary layers in viscous incompressible fluids

Le : 04/02/2025 de : 15:30 à : 16:30
When a fluid with small viscosity flows around an obstacle, a zone of small width is usually created in the vicinity of the wall. In this region, called « boundary layer », the velocity of the fluid changes abruptly. The study of fluid boundary layers has been an object of intense research over the past century. In this talk, we will focus on the stationary 2d Prandtl system, which was derived in 1905 by Ludwig Prandtl. We will present a review of results describing the flow in different regimes: without recirculation (i.e. when the tangential velocity close to the wall has the same sign as the velocity far from the obstacle); in the vicinity of the « separation point »; and with a recirculation bubble, i.e. when the tangential velocity changes sign.
Salle : A709

Colloquium du CEREMADE

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Intervenant : SERFATY Sylvia (Sorbonne Université)
Titre :
Coulomb systems

Le : 07/01/2025 de : 15:30 à : 16:30
We are interested in the statistical mechanics and dynamics of large systems of interacting Coulomb or Riesz particles. This is motivated by vortices, quantum and statistical mechanical models, approximation theory, and random matrix models. The mean-field behavior of these systems will be discussed, as well as their microscopic behavior as a function of temperature, which leads in particular to questions of crystallization.
Salle : A709

Colloquium du CEREMADE

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Intervenant : BUCUR Dorin (Université de Savoie)
Titre :
A free boundary approach of spectral optimization problems

Le : 03/12/2024 de : 15:30 à : 16:30
Why all the drums are round! This assertion is mathematically rephrased as follows: among all membranes with prescribed area, the one producing the lowest fundamental frequency has a circular shape. But what about the shapes of membranes with higher extremal frequencies? The general question in which are are interested concerns the relationship between the geometry of the domain and the spectrum of a differential operator defined on this domain. In this talk I will focus on the Neumann eigenvalues of the Laplace operator on domains of Euclidean space and on spheres. Together with a discussion about existence/non existence of optimal geometries and possible relaxation on densities, I will give some numerical approximations of the best geometries and densities. A particular attention will be given to the lowest two non trivial eigenvalues for which a full answer is given in any dimension of the Euclidean space. A surprising phenomenon occurs on spheres: while a complete answer can be given for the second eigenvalue, for the first one an unexplained phenomenon occurs. The results presented in this talk have been obtained in series of collaborations with R. Laugesen, A. Henrot, E. Martinet, M. Nahon and E. Oudet.
Salle : D207

Colloquium du CEREMADE

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Intervenant : DUVAL Céline (Sorbonne Université)
Titre :
Geometry of excursion sets: Computing the surface area from discretized points

Le : 05/11/2024 de : 15:30 à : 16:30
The excursion sets of a smooth random field carry information on the field that can be recovered using various geometric measures, such as the area or surface area of the excursion sets. After an introduction on these geometrical quantities, showing how they can be related to some parameters of the field, we focus on the problem of discretization. One never has access to the continuous observation of the excursion sets, but rather to observations at discrete points in space. In dimensions 2 and 3 and for specific regular grids, it has been reported that the usual estimate of the surface area of the excursion sets remains biased even when the grid becomes dense in the domain of observation. We study this limiting bias in any dimension and for polytopic tessellations. (Based on joint work with R. Cotsakis and E. Di Bernardino).
Salle : A709

Colloquium du CEREMADE

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Intervenant : BODINEAU Thierry (Ecole Polytechnique)
Titre :
A renormalisation group perspective on functional inequalities

Le : 01/10/2024 de : 15:30 à : 16:30
Functional inequalities provide information on the structure of a probability measure and on the relaxation of associated stochastic dynamics to equilibrium. In this talk, we will describe a multiscale analysis for decomposing high-dimensional measures into simpler structures and derive from it functional inequalities. The strategy is based on the renormalization group method used in statistical physics to study the distribution of interacting particle systems. We will also review related developments and in particular show that this decomposition of measures can be interpreted in terms of measure transport.
Salle : A709

Colloquium du CEREMADE

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Intervenant : MAïDA Mylène (Université de Lille)
Titre :
Mathematical aspects of 2D Yang-Mills theory

Le : 04/06/2024 de : 15:30 à : 16:30
In the fifties, Chen Ning Yang and Robert Mills made a major breakthrough in quantum field theory by extending the concept of gauge theory to non-abelian groups. Since then, the study of Yang-Mills theory has been a very active field of research both in mathematics and physics, in various directions. In particular, the theory on two-dimensional manifolds with gauge group U(N) or SU(N) has been studied as an (already very interesting) toy model, first by physicists in the nineties and then by mathematicians in the last two decades. If this talk, I will give an overview of the main mathematical results in this direction. Hopefully, if time allows, I will also explain in more details how the probabilistic study of well chosen random partitions allows us to give a reasonably simple rigorous proof of some topological expansions of the partition function of 2D Yang-Mills, predicted by physicists Gross and Taylor in the nineties (joint work with Thibaut Lemoine, Université de Lille).
Salle : C206

Colloquium du CEREMADE

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Intervenant : SANTAMBROGIO Filippo (Université Lyon 1)
Titre :
{Euclidean, metric, and Wasserstein} Gradient flows

Le : 07/05/2024 de : 15:30 à : 16:30
At the beginning of the presentation, I will explain what a gradient flow is in the simplest case: a curve in R^n solution of x'=-DF(x). In particular I will introduce a very natural time-discretization which is well-defined even when we replace the Euclidean space with a metric space, and I will briefly introduce some notions of gradient flow in a metric framework. We will then focus on the case where the metric space is the space of probability measures endowed with the Wasserstein distance, defined in terms of optimal transport. In this case we look for a mass distribution which evolves in time, and we can characterize it by PDEs. I will show how certain well-known diffusion PDEs (including the heat equation) fit into this framework.
Salle : A709

Colloquium du CEREMADE

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Intervenant : TóTH Bálint (University of Bristol and Alfréd Rényi Institute of Mathematics)
Titre :
The Random Lorentz Gas - Invariance Principle Beyond the Kinetic Time Scales

Le : 02/04/2024 de : 15:30 à : 16:30
Since the pioneering work of Paul and Tatiana Ehrenfest (1912) the deterministic (Hamiltonian) motion of a point-like particle exposed to the action of a collection of fixed, randomly located short range scatterers has been a much studied model of physical diffusion under fully deterministic (Hamiltonian) dynamics, with random initial conditions. This model of physical diffusion is known under the name of "random Lorentz gas" or "random wind-tree model". Celebrated milestones on the route to better mathematical understanding of this model of true physical diffusion are the Kinetic Limits for the tagged particle trajectory under the so-called Boltzmann-Grad (a.k.a. low density), or weak coupling approximations [Gallavotti (1970), Spohn (1978), Boldrighini-Bunimovich-Sinai (1982), respectively, Kesten-Papanicolaou (1980)]. Once the Kinetic Limits are established, under a second diffusive space-time scaling limit the central limit theorem (CLT) and invariance principle (IP) follows. However, the CLT/IP under bare diffusive space-time scaling (without first applying the kinetic approximations) remains a Holy Grail. In recent work we have obtained some intermediate results, partially interpolating between the two-steps-limit (first kinetic, then diffusive - as described above) and the bare-diffusive-limit (Holy Grail). We establish the Invariance Principle for the tagged particle trajectories under a joint kinetic+diffusive limiting procedure, performed simultaneously rather than successively, reaching significantly longer time scales than any earlier result. The main ingredient is a coupling of the Hamiltonian trajectory (with random initial conditions) and an approximate Markovized version of the motion, and probabilistic and geometric controls on the efficiency of this coupling. The Holy Grail remains, however, beyond reach. In the colloquium talk I will present a concise historic survey of the problems and some insight regarding the main ideas of the more recent work.
Salle : A709

Colloquium du CEREMADE

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Intervenant : CASTILLO Ismaël (Sorbonne Université)
Titre :
A tale of heavy tails

Le : 05/03/2024 de : 15:30 à : 16:30
Dans cet exposé, j’expliquerai l’utilité de lois de probabilité à queues lourdes en statistique bayésienne en tant que lois a priori sur des aspects d’un objet fonctionnel à estimer, par exemple les coefficients d’une fonction sur une base. En partant d’une comparaison avec les processus gaussiens, nous verrons que, de façon surprenante au premier abord, des propriétés automatiques d’adaptation à la régularité de la fonction inconnue sont obtenues avec certaines lois à queues lourdes. Nous présenterons aussi des propriétés d’adaptation à des structures cachées telles que les structures de composition pour des réseaux de neurones profonds dont les poids sont à queues lourdes. Nous expliquerons certains avantages du point de vue algorithmique, comme l’absence de nécessité d’échantillonnage sur les (hyper-)paramètres de régularité ou de structure.
Salle : A709

Colloquium du CEREMADE

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Intervenant : CAPUTO Pietro (Università Roma Tre)
Titre :
Navigating Networks by Random Walks
https://www.ceremade.dauphine.fr/~salez/colloquium.html
Le : 05/12/2023 de : 15:30 à : 16:30
In the analysis of a large, complex network, exploration via random walks is often the most effective strategy available. How long does it take to traverse the entire network ? Does the walk achieve a stable equilibrium, and if so, how long is the journey to reach it ? And crucially, what does this steady state look like ? The answers to these questions are key to understanding a network's structural features and provide valuable insights for ranking systems and search algorithms. Developing a comprehensive mathematical treatment of these issues for real-world networks poses a formidable task, and a rigorous analysis has only recently started in the context of relatively simple models. In this lecture, we illustrate some promising preliminary progress in the setting of directed networks obtained from simple random graph models. The discussion includes scenarios where the walker experiences regeneration events such as teleportation, as in PageRank algorithms, or resampling of the underlying graph, as in a dynamically evolving network.
Salle : C206

Colloquium du CEREMADE

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Intervenant : MAURY Bertrand (Université Paris Sud)
Titre :
Sciences sociales et flots de gradient

Le : 07/11/2023 de : 15:30 à : 16:30
Un grand nombre de phénomènes physiques recèle une structure de flot de gradient, ou de de système hamiltonien (que l’on peut voir comme une version inertielle du flot de gradient). Cela signifie qu’il existe une fonction sous-jacente des variables d’état (de positions s’il s’agit par exemple de particules), de type énergie potentielle dans un contexte mécanique, qui conditionne l’évolution du système. Dans la version non inertielle, l’état du système « glisse » suivant la ligne de plus grande pente de cette fonction, qui conditionne donc entièrement le comportement global du système. Nous nous demanderons si certains phénomènes impliquant des entités pensantes et dotées de capacités cognitives, et qui peuvent a priori se modéliser par des équations formellement proches de celles de la physique des entités passives, présentent cette structure. Nous tâcherons de préciser ce qui peut expliquer qu’ils s’en écartent, légèrement ou plus franchement, et nous explorerons les conséquences que cela peut avoir sur le comportement des solutions. Nous illustrerons ces considération générales dans le domaine de la propagation d’opinion sur réseaux sociaux, et sur les mouvements de foules ou de véhicules.
Salle : A709

Colloquium du CEREMADE

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Intervenant : LELARGE Marc (INRIA et Ecole Polytechnique)
Titre :
Mathématiques et Intelligence Artificielle

Le : 03/10/2023 de : 15:30 à : 16:30
Les principaux algorithmes d'apprentissage statistique reposent sur des fondements mathématiques variés tels que les statistiques, l'optimisation, les probabilités... En sera-t-il de même pour l'Intelligence Artificielle? Dans cette présentation, je présenterai quelques exemples récents d'interactions entre apprentissage machine et mathématiques.
Salle : A709

Colloquium du CEREMADE

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Intervenant : YOUSSEF Pierre (NYUAD)
Titre :
Phenomena in High Dimensions

Le : 09/06/2023 de : 14:00 à : 15:00
How do high dimensional convex bodies look like? How does the spectrum of a random matrix behave as its dimension grows? How does the connectivity of a graph evolve as its size grows? In this talk, I will try to discuss some phenomena that arise as the ``dimension'' grows in several related contexts.
Salle : A311

Colloquium du CEREMADE

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Intervenant : ANANTHARAMAN Nalini (Université de Strasbourg)
Titre :
Principes d'incertitude, inégalités d'incertitude

Le : 04/04/2023 de : 15:30 à : 16:30
Je présenterai un panorama (non exhaustif) de résultats classiques et moins classiques se rattachant au principe d'incertitude : il n'est pas possible simultanément pour une fonction et sa transformée de Fourier d'être concentrées sur des ensembles trop petits. Je montrerai ensuite quelques liens avec un de mes sujets de prédilection, l'ergodicité quantique.
Salle : A709

Colloquium du CEREMADE

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Intervenant : ROSENBAUM Mathieu (Ecole Polytechnique)
Titre :
Des tremblements de terre à la compréhension des risques financiers via le football: autour de la modélisation statistique par les processus de Hawkes.

Le : 03/01/2023 de : 15:30 à : 16:30
Le but de cet exposé est d'illustrer la pertinence et la flexibilité des processus de Hawkes comme outil de modélisation pour de nombreux problèmes statistiques. Nous montrerons notamment comment ceux-ci ont pu permettre de revisiter notre compréhension de la dynamique des risques en finance et de créer de nouvelles approches de gestion quantitative plus efficaces. Nous évoquerons aussi leur utilisation pour l'analyse de données sportives dans le cadre de la construction de métriques de performance.
Salle : A709

Colloquium du CEREMADE

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Intervenant : STAUFFER Alexandre (University of Bath)
Titre :
Random growth processes: dendritic formation and competition

Le : 06/12/2022 de : 15:30 à : 16:30
During the 80's, several growth processes were introduced in the physics literature with the goal of providing a simple and tractable model of dendritic growth. Such phenomenon is observed in several different contexts in nature, including the growth of bacteria under starvation, crystal dendrite, dielectric breakdown, and electrodeposition, and is characterized by the formation of very ramified, fractal-like structures. Almost four decades later, we still encounter tremendous mathematical challenges in studying the geometric and dynamic properties of such processes. In this talk, I will survey the developments in this field, giving emphasis to recent results in the classical model of multi-particle diffusion limited aggregation (MDLA). I will also introduce and discuss a new growth process, which can be viewed as a model for the growth of two species that compete for space. This new process has been recently used to analyze MDLA and other models.
Salle : A709

Colloquium du CEREMADE

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Intervenant : VAN DEN BOSCH Hanne (Universidad de Chile)
Titre :
Solitary waves in Nonlinear Dirac Equations
Colloquium (suivi d'un pot)
Le : 04/10/2022 de : 15h30: à : 16h30:
In nonlinear dispersive equations, the competition between the linear dispersion and nonlinear self-interaction gives rise to localized solutions that maintain their shape as time evolves. Famous examples of this are the solitons in the Korteweg-de Vries equation or solitary waves for the nonlinear Schrödinger equation. In the latter case, there is a beautiful theory to determine if small perturbations in the initial condition remain close to a solitary wave as time evolves, or on the contrary, may completely destroy its shape. In many types of Nonlinear Dirac equations, these special solutions exist as well, but very little is known rigorously about their properties. During the colloquium, I will present this problem and compare the results for the Schrödinger case with those for their Dirac analogues.
Salle : A709

Colloquium du CEREMADE

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Intervenant : LOSS Michael (Georgia Institute of Technology)
Titre :
Optimal criteria for magnetic fields that bind electrons

Le : 30/05/2022 de : 14:30 à : 15:30
This talk will provide a basic introduction to the three dimensional Dirac equation that describes an electron interacting with a magnetic field. Over the years a lot of work has gone into constructing zero energy solutions, also known as zero modes, for said equation. In this talk I will explain the importance of zero modes and will show how they relate to the stability of the hydrogen atom. After presenting explicit examples, I will state necessary conditions for the magnetic field so that zero modes exist. Here, of particular interest is a sharp inequality that is optimized by a magnetic field whose field lines are interlinking circles. This pattern results from pulling back the Hopf fibration on the three sphere to three dimensional space using the stereographic projection.
Salle : A709

Colloquium du CEREMADE

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Intervenant : GALLAGHER Isabelle (Ecole Normale Supérieure de Paris)
Titre :
Sur la dynamique des gaz dilués

Le : 05/04/2022 de : 15:30 à : 16:30
L'évolution d'un gaz peut être décrite par différents modèles selon l'échelle d'observation. Une question naturelle, soulevée par Hilbert dans son sixième problème, est de savoir si ces modèles fournissent des prédictions cohérentes. Dans le cas des gaz de sphères dures, Lanford a montré en 1974 que l'équation de Boltzmann apparaît comme une loi des grands nombres dans la limite de faible densité, au moins pour des temps très courts. Dans cet exposé nous présenterons le résultat de Lanford, et quelques extensions récentes permettant de comprendre les fluctuations et les grandes déviations autour de l'équation de Boltzmann.
Salle : A709

Colloquium du CEREMADE

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Intervenant : GASSIAT Elisabeth (Université Paris Saclay)
Titre :
Quand la dépendance ouvre des possibles

Le : 07/12/2021 de : 15:30 à : 16:30
C’est une idée générale et vague qu’un peu de structure dans les données renforce les capacités d’apprentissage. J’explorerai cette idée en introduisant de la dépendance dans la modélisation et en m’appuyant plus particulièrement sur le clustering. Le clustering est une méthode de classification non super-visée qui cherche à regrouper les données en sous-populations signifiantes, dites « clusters ». Mais comment définir un cluster ? Je présenterai les modèles à variable latente, et montrerai comment une structure de dépendance markovienne des variables latentes permet de s’affranchir de toute hypothèse restrictive sur les clusters. Je présenterai aussi d’autres exemples de modèles où la dépendance permet de résoudre des questions qui ne peuvent l’être dans le cadre indépendant.
Salle : A709

Colloquium du CEREMADE

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Intervenant : CURIEN Nicolas (Université Paris Saclay)
Titre :
Surfaces aléatoires en tous genres

Le : 05/10/2021 de : 15:30 à : 16:30
Qu'est-ce qu'une surface aléatoire ? Cette question mathématique naturelle intervient également en physique, dans la théorie de la gravité quantique en dimension 2. Bien sûr la réponse dépend de la loi de probabilité choisie. Nous verrons quelques propriétés des surfaces discrètes aléatoires (connues sous le nom de cartes aléatoires). Le cas planaire (genre 0) est maintenant bien étudié en probabilités avec notamment les travaux de Le Gall, Miermont, Sheffield et Miller sur la carte brownienne. Mais quid du genre plus grand ? Nous esquisserons aussi quelques réponses pour les surfaces aléatoires "continues". L'exposé comportera beaucoup, beaucoup d'images.
Salle : A709