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## Nos 50 dernières publications

• 28 June 2022

Frédéric Jaeck

• 27 June 2022

We prove new vanishing results on the growth of higher torsion homologies for suitable arithmetic lattices, Artin groups and mapping class groups. The growth is understood along Farber sequences, in particular, along residual chains. For principal congruence subgroups, we also obtain strong asymptotic bounds for the torsion growth. As a central tool, we introduce a quantitative homotopical method called effective rebuilding. This constructs small classifying spaces of finite index subgroups, at the same time controlling the complexity of the homotopy. The method easily applies to free abelian groups and then extends recursively to a wide class of residually finite groups.

Miklos Abert, Nicolas Bergeron, Mikolaj Fraczyk, Damien Gaboriau

• 23 June 2022

This study focuses on ocean waves impacting the Moorea Island in French Polynesia, where coral reefs play an essential role in the biodiversity and protection of habitations. We investigate how the innovative SWIM instrument of the CFOSAT satellite enables to document on a multi-annual basis, the spectral properties of ocean waves reaching the coasts of the Moorea Island. Our analysis is based on comparisons with in situ measurements (wave gauges deployed on the outer slope of the coral reef), and with other satellite observations (altimeter, SAR). Accounting for local masking effects, we show that SWIM provides relevant information on short swell or wind waves, which is missed by the SAR observations, in particular in high sea-state conditions, owing to the dominant propagation direction being close to the azimuth. We, nevertheless, also find that wave properties in low sea-state conditions are better documented by SAR than by SWIM. Such results are important to accurately measure and predict the wave conditions which fragilize the coral reefs and to evaluate the impact of extreme events on tropical islands and coral reefs.

Ludivine Oruba, Danièle Hauser, Serge Planes, Emmanuel Dormy

• 19 May 2022

In this paper, we study the Landau equation under the Navier-Stokes scaling in the torus for hard and moderately soft potentials. More precisely, we investigate the Cauchy theory in a perturbative framework and establish some new short time regularization estimates for our rescaled nonlinear Landau equation. These estimates are quantified in time and optimal, indeed, we obtain the instantaneous expected anisotropic gain of regularity (see [53] for the corresponding hypoelliptic estimates on the linearized Landau collision operator). Moreover, the estimates giving the gain of regularity in the velocity variable are uniform in the Knudsen number. Intertwining these new estimates on the Landau equation with estimates on the Navier-Stokes-Fourier system, we are then able to obtain a result of strong convergence towards this fluid system.

Kleber Carrapatoso, Mohamad Rachid, Isabelle Tristani

• 7 May 2022

Hörmander's propagation of singularities theorem does not fully describe the propagation of singularities in subelliptic wave equations, due to the existence of doubly characteristic points. In the present work, building upon a visionary conference paper by R. Melrose \cite{Mel86}, we prove that singularities of subelliptic wave equations only propagate along null-bicharacteristics and abnormal extremals, which are well-known curves in optimal control theory. As a consequence, we characterize the singular support of subelliptic wave kernels outside the diagonal. These results show that abnormal extremals play an important role in the classical-quantum correspondence between sub-Riemannian geometry and sub-Laplacians.

Cyril Letrouit

• 7 May 2022

In [7], a cluster expansion method has been developed to study the fluctuations of the hard sphere dynamics around the Boltzmann equation. This method provides a precise control on the exponential moments of the empirical measure, from which the fluctuating Boltzmann equation and large deviation estimates have been deduced. The cluster expansion in [7] was implemented at the level of the BBGKY hierarchy, which is a standard tool to investigate the deterministic dynamics [11]. In this paper, we introduce an alternative approach, in which the cluster expansion is applied directly on real trajectories of the particle system. This offers a fresh perspective on the study of the hard sphere dynamics in the low density limit, allowing to recover the results obtained in [7], and also to describe the actual clustering of particle trajectories.

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella

• 5 May 2022

We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop Kähler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin-Abreu-Donaldson geometry of toric varieties. Namely we associate convex functions with hermitian metrics on line bundles, and express the curvature form in terms of this function, as well as the corresponding Monge-Ampère volume form and scalar curvature. We then provide an expression for the Mabuchi functional and derive as an application a combinatorial sufficient condition of properness similar obtained by Li, Zhou and Zhu on group compactifications.

Thibaut Delcroix

• 5 May 2022

This thesis proposes theoretical and numerical contributions to perform machine learning and statistics over the space of probability distributions. In a first part, we introduce a new class of neural network architectures to process probability measures in their Lagrangian form (obtained by sampling) as both inputs and outputs, which is characterized by robustness and universal approximation properties. We show that this framework can be adapted to perform regression on probability measure inputs, with customized invariance requirements, in a way that preserves its robustness and approximation capabilities. This method is proven to be of interest to design expressive, adaptable summaries of datasets referred to as “meta-features”, in the context of automated machine learning. In a second part, we demonstrate that the resort to entropy eases the computation of conditional multivariate quantiles. We introduce the regularized vector quantile regression problem, provide a scalable algorithm to compute multivariate quantiles and show that it benefits from desirable asymptotic properties.

Gwendoline de Bie

• 4 May 2022

The notion of propagation of chaos for large systems of interacting particles originates in statistical physics and has recently become a central notion in many areas of applied mathematics. The present review describes old and new methods as well as several important results in the field. The models considered include the McKean-Vlasov diffusion, the mean-field jump models and the Boltzmann models. The first part of this review is an introduction to modelling aspects of stochastic particle systems and to the notion of propagation of chaos. The second part presents concrete applications and a more detailed study of some of the important models in the field.

Louis-Pierre Chaintron, Antoine Diez

• 4 May 2022

The notion of propagation of chaos for large systems of interacting particles originates in statistical physics and has recently become a central notion in many areas of applied mathematics. The present review describes old and new methods as well as several important results in the field. The models considered include the McKean-Vlasov diffusion, the mean-field jump models and the Boltzmann models. The first part of this review is an introduction to modelling aspects of stochastic particle systems and to the notion of propagation of chaos. The second part presents concrete applications and a more detailed study of some of the important models in the field.

Louis-Pierre Chaintron, Antoine Diez

• 21 April 2022

After observing that some constructions and results in the p-adic Langlands programme are somehow independent from p, we formulate the hypothesis that this astonishing uniformity could be explained by a 1-adic Langlands correspondence.

Xavier Caruso, Agnès David, Ariane Mézard

• 20 April 2022

We investigate here the interactions of waves governed by a Boussinesq system with a partially immersed body allowed to move freely in the vertical direction. We show that the whole system of equations can be reduced to a transmission problem for the Boussinesq equations with transmission conditions given in terms of the vertical displacement of the object and of the average horizontal discharge beneath it; these two quantities are in turn determined by two nonlinear ODEs with forcing terms coming from the exterior wave-field. Understanding the dispersive contribution to the added mass phenomenon allows us to solve these equations, and a new dispersive hidden regularity effect is used to derive uniform estimates with respect to the dispersive parameter. We then derive an abstract general Cummins equation describing the motion of the solid in the return to equilibrium problem and show that it takes an explicit simple form in two cases, namely, the nonlinear non dispersive and the linear dispersive cases; we show in particular that the decay rate towards equilibrium is much smaller in the presence of dispersion. The latter situation also involves an initial boundary value problem for a nonlocal scalar equation that has an interest of its own and for which we consequently provide a general analysis.

Geoffrey Beck, David Lannes

• 3 April 2022

This work deals with the interaction of waves governed by a Boussinesq system with some floating structures. The full system can be reduced to coupled boundary problems for the Boussinesq equations with boundary conditions given in terms of the vertical displacement of the objects, the average horizontal discharge beneath it and the traces of the water-column. The latter quantities are determined by nonlinear ODEs with forcing terms coming from the exterior wavefield.

Geoffrey Beck, David Lannes, Lisl Weynans

• 30 March 2022

In this work we construct an efficient numerical method to solve 3D Maxwell's equations in coaxial cables. Our strategy is based upon an hybrid explicit-implicit time discretization combined with edge elements on prisms and numerical quadrature. One of the objective is to validate numerically generalized Telegrapher's models that are used to simplify the 3D Maxwell equations into a 1D problem. This is the object of the second part of the article.

Akram Beni-Hamad, Geoffrey Beck, Sébastien Imperiale, Patrick Joly

• 25 March 2022

We prove two results concerning the nodal sets of eigenfunctions of sub-Laplacians. The first one asserts the validity in this setting of Courant's theorem on the number of nodal domains of eigenfunctions. The second one is the C/\sqrt{\lambda} density (with respect to the sub-Riemannian distance) of the nodal sets of eigenfunctions with eigenvalue \lambda.

Cyril Letrouit

• 17 March 2022

Cette thèse propose des avancées théoriques et algorithmiques pour les relaxations semidéfinies positives et leurs applications en science des données. Ces relaxations, dites de Lasserre, fondées sur la substitution aux mesures boréliennes de leurs moments trigonométriques, permettent de résoudre des problèmes de super-résolution sans discrétisation spatiale, mais nécessitent en contrepartie la résolution de problèmes d’optimisation convexe de grande taille. Les contributions de cette thèse montrent comment faire passer ces méthodes à l’échelle en exploitant certaines propriétés et invariances des problèmes d’imagerie. Nous proposons dans un premier temps une nouvelle méthode pour la reconstruction de mesures continues à partir d’un nombre fini de leurs moments, reposant sur un algorithme de codiagonalisation approchée. Nous étudions ensuite le problème de super-résolution, sous sa forme variationnelle appelée BLASSO, et son approximation par la hiérarchie de Lasserre. Une étape préalable de projection spectrale de l’opérateur d’acquisition rend possible cette approximation et permet également son implémentation efficace via un nouvel algorithme, le Fourier-based Frank- Wolfe (FFW), tirant profit de la structure convolutive et de faible rang des matrices impliquées. Nous appliquons notre méthode sur des données de microscopie par fluorescence. Enfin, combinant la reconstruction de mesures continues avec l’implémentation rapide de FFW, nous employons les hiérarchies de Lasserre afin d’approcher le transport optimal entre deux mesures, qui peut également être couplé avec le Blasso.

Paul Catala

• 10 March 2022

Distributed computing offers a high degree of flexibility to accommodate modern learning constraints and the ever increasing size of datasets involved in massive data issues. Drawing inspiration from the theory of distributed computation models developed in the context of gradient-type optimization algorithms, we present a consensus-based asynchronous distributed approach for nonparametric online regression and analyze some of its asymptotic properties. Substantial numerical evidence involving up to 28 parallel processors is provided on synthetic datasets to assess the excellent performance of our method, both in terms of computation time and prediction accuracy.

Gérard Biau, Ryad Zenine

• 5 March 2022

The evolution of a gas can be described by different models depending on the observation scale. A natural question, raised by Hilbert in his sixth problem, is whether these models provide consistent predictions. In particular, for rarefied gases, it is expected that continuum laws of kinetic theory can be obtained directly from molecular dynamics governed by the fundamental principles of mechanics. In the case of hard sphere gases, Lanford showed that the Boltzmann equation emerges as the law of large numbers in the low density limit, at least for very short times. The goal of this survey is to present recent progress in the understanding of this limiting process, providing a complete statistical description.

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella

• 23 February 2022

We construct measures on definable sets in $e$-free perfect PAC fields, as well as on perfect PAC fields whose absolute Galois groups are free pro-$p$ of finite rank. We deduce the definable amenability of all groups definable in such fields. As a corollary, we additionally prove the definable amenability of all groups definable in perfect $\omega$-free PAC fields via ultralimit measures.

Zoé Chatzidakis, Nicholas Ramsey

• 23 February 2022

The three-dimensional, homogeneous, incompressible Navier-Stokes equations are studied in the absence of viscosity in one direction. It is shown that there are arbitrarily large initial data generating a unique global solution, the main feature of which is that they are slowly varying in the direction where viscosity is missing. The difficulty arises from the complete absence of a regularising effect in this direction. The special structure of the nonlinear term, joint with the divergence-free condition on the velocity field, is crucial in obtaining the result.

Isabelle Gallagher, Alexandre yotopoulos

• 12 February 2022

The difficulties associated with network connectivity, unreliable channels, and city environment characteristics make data dissemination task in vehicular urban networks a real challenge. Recently, some interesting solutions have been proposed to perform data dissemination in this environment. Starting from the analysis of these solutions, we present a new dissemination protocol named DHVN (Dissemination protocol for Heterogeneous Cooperative Vehicular Networks) that considers: (i) roads topology, (ii) network connectivity and possible partitioning in case of low traffic density, and (iii) heterogeneous communication capabilities of the vehicles. We compare our protocol to other dissemination protocols and analyze its performances using NS-3 simulator [1]. Performance studies show interesting DHVN compared to existing solutions. Indeed, DHVN is able to provide a low end-to-end delay, a high delivery ratio and a minimum bandwidth usage since only a limited number of vehicles are involved in the broadcast scheme.

Sara Mehar, Sidi Mohammed Senouci, Guillaume Remy

• 1 February 2022

In personalized Federated Learning, each member of a potentially large set of agents aims to train a model minimizing its loss function averaged over its local data distribution. We study this problem under the lens of stochastic optimization. Specifically, we introduce informationtheoretic lower bounds on the number of samples required from all agents to approximately minimize the generalization error of a fixed agent. We then provide strategies matching these lower bounds, in the all-for-one and all-for-all settings where respectively one or all agents desire to minimize their own local function. Our strategies are based on a gradient filtering approach: provided prior knowledge on some notions of distances or discrepancies between local data distributions or functions, a given agent filters and aggregates stochastic gradients received from other agents, in order to achieve an optimal bias-variance trade-off.

Mathieu Even, Laurent Massoulié, Kévin Scaman

• 28 January 2022

We consider the Bernoulli percolation model in a finite box and we introduce an automatic control of the percolation probability, which is a function of the percolation configuration. For a suitable choice of this automatic control, the model is self-critical, i.e., the percolation probability converges to the critical point pc when the size of the box tends to infinity. We study here three simple examples of such models, involving the size of the largest cluster, the number of vertices connected to the boundary of the box, or the distribution of the cluster sizes. Along the way, we prove a general geometric inequality for subgraphs of Z d , which is of independent interest.

Raphaël Cerf, Nicolas Forien

• 26 January 2022

We characterize all random point measures which are in a certain sense stable under the action of branching. Denoting by $\circledast$ the branching convolution operation introduced by Bertoin and Mallein (2019), and by $\mathcal{Z}$ the law of a random point measure on the real line, we are interested in solutions to the fixed point equation $\mathcal E = \mathcal{Z} \circledast \mathcal E,$ with $\mathcal E$ a random point measure distribution. Under suitable assumptions, we characterize all solutions of this equation as shifted decorated Poisson point processes with a uniquely defined shift.

Pascal Maillard, Bastien Mallein

• 26 January 2022

The original problem of group testing consists in the identification of defective items in a collection, by applying tests on groups of items that detect the presence of at least one defective item in the group. The aim is then to identify all defective items of the collection with as few tests as possible. This problem is relevant in several fields, among which biology and computer sciences. It recently gained attraction as a potential tool to solve a shortage of COVID-19 test kits, in particular for RT-qPCR. However, the problem of group testing is not an exact match to this implementation. Indeed, contrarily to the original problem, PCR testing employed for the detection of COVID-19 returns more than a simple binary contaminated/non-contaminated value when applied to a group of samples collected on different individuals. It gives a real value representing the viral load in the sample instead. We study here the use that can be made of this extra piece of information to construct a one-stage pool testing algorithms on an idealize version of this model. We show that under the right conditions, the total number of tests needed to detect contaminated samples diminishes drastically.

Emilien Joly, Bastien Mallein

• 26 January 2022

Consider a weighted branching process generated by a point process on $[0,1]$, whose atoms sum up to one. Then the weights of all individuals in any given generation sum up to one, as well. We define a nested occupancy scheme in random environment as the sequence of balls-in-boxes schemes (with random probabilities) in which boxes of the $j$th level, $j=1,2,\ldots$ are identified with the $j$th generation individuals and the hitting probabilities of boxes are identified with the corresponding weights. The collection of balls is the same for all generations, and each ball starts at the root and moves along the tree of the weighted branching process according to the following rule: transition from a mother box to a daughter box occurs with probability given by the ratio of the daughter and mother weights. Assuming that there are $n$ balls, we give a full classification of regimes of the a.s.\ convergence for the number of occupied (ever hit) boxes in the $j$th level, properly normalized, as $n$ and $j=j_n$ grow to $\infty$. Here, $(j_n)_{n\in\mathbb N}$ is a sequence of positive numbers growing proportionally to $\log n$. We call such levels late, for the nested occupancy scheme gets extinct in the levels which grow super-logarithmically in $n$ in the sense that each occupied box contains one ball. Also, in some regimes we prove the strong laws of large numbers (a) for the number of the $j$th level boxes which contain at least $k$ balls, $k\geq 2$ and (b) under the assumption that the mean number of the first level boxes is finite, for the number of empty boxes in the $j$th level.

Alexander Iksanov, Bastien Mallein

• 26 January 2022

Consider a weighted branching process generated by a point process on $[0,1]$, whose atoms sum up to one. Then the weights of all individuals in any given generation sum up to one, as well. We define a nested occupancy scheme in random environment as the sequence of balls-in-boxes schemes (with random probabilities) in which boxes of the $j$th level, $j=1,2,\ldots$ are identified with the $j$th generation individuals and the hitting probabilities of boxes are identified with the corresponding weights. The collection of balls is the same for all generations, and each ball starts at the root and moves along the tree of the weighted branching process according to the following rule: transition from a mother box to a daughter box occurs with probability given by the ratio of the daughter and mother weights. Assuming that there are $n$ balls, we give a full classification of regimes of the a.s.\ convergence for the number of occupied (ever hit) boxes in the $j$th level, properly normalized, as $n$ and $j=j_n$ grow to $\infty$. Here, $(j_n)_{n\in\mathbb N}$ is a sequence of positive numbers growing proportionally to $\log n$. We call such levels late, for the nested occupancy scheme gets extinct in the levels which grow super-logarithmically in $n$ in the sense that each occupied box contains one ball. Also, in some regimes we prove the strong laws of large numbers (a) for the number of the $j$th level boxes which contain at least $k$ balls, $k\geq 2$ and (b) under the assumption that the mean number of the first level boxes is finite, for the number of empty boxes in the $j$th level.

Romain Biard, Bastien Mallein, Landy Rabehasaina

• 17 January 2022

In the study of Hamiltonian systems on cotangent bundles, it is natural to perturb Hamiltoni-ans by adding potentials (functions depending only on the base point). This led to the definition of Mañé genericity: a property is generic if, given a Hamiltonian H, the set of potentials u such that H + u satisfies the property is generic. This notion is mostly used in the context of Hamiltonians which are convex in p, in the sense that ∂ 2 pp H is positive definite at each points. We will also restrict our study to this situation. There is a close relation between perturbations of Hamiltonians by a small additive potential and perturbations by a positive factor close to one. Indeed, the Hamiltonians H + u and H/(1 − u) have the same level one energy surface, hence their dynamics on this energy surface are reparametrisation of each other, this is the Maupertuis principle. This remark is particularly relevant when H is homogeneous in the fibers (which corresponds to Finsler metrics) or even fiberwise quadratic (which corresponds to Riemannian metrics). In these cases, perturbations by potentials of the Hamiltonian correspond, up to parametrisation, to conformal perturbations of the metric. One of the widely studied aspects is to understand to what extent the return map associated to a periodic orbit can be perturbed by adding a small potential. This kind of question depend strongly on the context in which they are posed. Some of the most studied contexts are, in increasing order of difficulty, perturbations of general vector fields, perturbations of Hamiltonian systems inside the class of Hamiltonian systems, perturbations of Riemannian metrics inside the class of Riemannian metrics, Mañé perturbations of convex Hamiltonians. It is for example well-known that each vector field can be perturbed to a vector field with only hyperbolic periodic orbits, this is part of the Kupka-Smale theorem, see [5, 13]. There is no such result in the context of Hamiltonian vector fields, but it remains true that each Hamiltonian can be perturbed to a Hamiltonian with only non-degenerate periodic orbits (including the iterated ones), see [11, 12]. The same result is true in the context of Riemannian metrics: every Riemannian metric can be perturbed to a Riemannian metric with only non-degenerate closed geodesics, this is the bumpy metric theorem, see [4, 2, 1]. The question was investigated only much more recently in the context of Mañé perturbations of convex Hamiltonians, see [9, 10]. It is proved in [10] that the same result holds : If H is a convex Hamiltonian and a is a regular value of H, then there exist arbitrarily small potentials u such that all periodic orbits (including iterated ones) of H + u at energy a are non-degenerate. The proof given in [10] is actually rather similar to the ones given in papers on the perturbations of Riemannian metrics. In all these proofs, it is very useful to work

Shahriar Aslani, Patrick Bernard

• 17 January 2022

We prove a bumpy metric theorem in the sense of Ma\~{n}e for non-convex Hamiltonians that are satisfying a certain geometric property.

Shahriar Aslani, Patrick Bernard

• 12 January 2022

We study a hard sphere gas at equilibrium, and prove that in the low density limit, the fluctuations converge to a Gaussian process governed by the fluctuating Boltzmann equation. This result holds for arbitrarily long times. The method of proof builds upon the weak convergence method introduced in the companion paper [8] which is improved by considering clusters of pseudo-trajectories as in [7].

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella

• 11 January 2022

Throughout this PhD thesis we will study two probabilistic objects, Gaussian multiplicative chaos (GMC) measures and Liouville conformal field theory (LCFT). The GMC measures were first introduced by Kahane in 1985 and have grown into an extremely important field of probability theory and mathematical physics. Very recently GMC has been used to give a probabilistic definition of the correlation functions of LCFT, a theory that first appeared in Polyakov's 1981 seminal work, "Quantum geometry of bosonic strings". Once the connection between GMC and LCFT is established, one can hope to translate the techniques of conformal field theory in a probabilistic framework to perform exact computations on the GMC measures. We start from the BPZ equations for LCFT, introduced by Belavin, Polyakov and Zamolodchikov in 1983. The mechanism of these equations is studied in the last part of this thesis and we prove the higher order BPZ equations with a general formalism. Following the probabilistic methods established by Kupiainen-Rhodes-Vargas for the resolution of the BPZ equations and after overcoming several major difficulties, we obtain non trivial relations for some fundamental objects of LCFT. More precisely, we prove the exact formulas for all the four structure constants of LCFT on the disk with null cosmological constant in the bulk, one of which was solved by Remy in 2017. As a special case, we find the distribution of the total mass of GMC on the interval with log-singularities put on both ends, a conjecture that has been independently predicted by Ostrovsky and by Fyodorov, Le Doussal, and Rosso in 2009. Another direct consequence is the law of the total mass of GMC on the unit circle with a log-singularity, conjectured by Ostrovsky in 2016.

Tunan Zhu

• 8 January 2022

In this work, we prove the convergence of strong solutions of the Boltzman equation, for initial data having polynomial decay in the velocity variable, towards those of the incompressible Navier-Stokes-Fourier system. We show in particular that the solutions of the rescaled Boltzmann equation do not blow up before their hydrodynamic limit does. This is made possible by adapting the strategy from [7] of writing the solution to the Boltzmann equation as the sum a part with polynomial decay and a second one with Gaussian decay. The Gaussian part is treated with an approach reminiscent of the one from [17].

Pierre Gervais

• 30 December 2021

For dimensions $n \geq 3$, we classify singular solutions to the generalized Liouville equation $(-\Delta)^{n/2} u = e^{nu}$ on $\mathbb{R}^n \setminus \{0\}$ with the finite integral condition $\int_{\mathbb{R}^n} e^{nu} < \infty$ in terms of their behavior at $0$ and $\infty$. These solutions correspond to metrics of constant $Q$-curvature which are singular in the origin. Conversely, we give an optimal existence result for radial solutions. This extends some recent results on solutions with singularities of logarithmic type to allow for singularities of arbitrary order. As a key tool to the existence result, we derive a new weighted Moser--Trudinger inequality for radial functions.

Tobias König, Paul Laurain

• 17 December 2021

We prove an existence result for the principal-agent problem with adverse selection under general assumptions on preferences and allo-cation spaces. Instead of assuming that the allocation space is ﬁnite-dimensional or compact, we consider a more general coercivity condi-tion which takes into account the principal’s cost and the agents’ pref-erences. Our existence proof is simple and ﬂexible enough to adapt to partial participation models as well as to the case of type-dependent budget constraints.

Guillaume Carlier, Kelvin Shuangjian Zhang

• 2 December 2021

We study the rapid decay property and polynomial growth for duals of bicrossed products coming from a matched pair of a discrete group and a compact group.

Pierre Fima, Hua Wang

• 30 November 2021

In this paper, we give an overview of the results established in [3] which provides the first rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres in 3D. In particular, we obtain a new system of hydrodynamic equations describing granular flows and prove existence of classical solutions to the aforementioned system. One of the main issue is to identify the correct relation between the restitution coefficient (which quantifies the rate of energy loss at the microscopic level) and the Knudsen number which allows us to obtain non trivial hydrodynamic behavior. In such a regime, we construct strong solutions to the inelastic Boltzmann equation, near thermal equilibrium whose role is played by the so-called homogeneous cooling state. We prove then the uniform exponential stability with respect to the Knudsen number of such solutions, using a spectral analysis of the linearized problem combined with technical a priori nonlinear estimates. Finally, we prove that such solutions converge, in a specific weak sense, towards some hydrodynamic limit that depends on time and space variables only through macroscopic quantities that satisfy a suitable modification of the incompressible Navier-Stokes-Fourier system.

Ricardo J Alonso, Bertrand Lods, Isabelle Tristani

• 26 November 2021

We present an elementary approach to prove restriction theorems for particular surfaces for which the Tomas-Stein theorem does not apply, which in turn provide short proofs for well-known Strichartz estimates for associated PDEs. The method consists in applying simple restriction theorems to the level sets of the surfaces studied (for instance, spheres in the case of the cone) and then integrating the inequalities over all level sets. This allows for a different proof of sharp estimates for the wave equation, and of some estimates related to the Euler equations in the rotational framework.

Corentin Gentil, Côme Tabary

• 19 November 2021

We investigate the behavior of the spectrum of the continuous Anderson Hamiltonian $\mathcal{H}_L$, with white noise potential, on a segment whose size $L$ is sent to infinity. We zoom around energy levels $E$ either of order $1$ (Bulk regime) or of order $1\ll E \ll L$ (Crossover regime). We show that the point process of (appropriately rescaled) eigenvalues and centers of mass converge to a Poisson point process. We also prove exponential localization of the eigenfunctions at an explicit rate. In addition, we show that the eigenfunctions converge to well-identified limits: in the Crossover regime, these limits are universal. Combined with the results of our companion paper arXiv:2102.05393, this identifies completely the transition between the localized and delocalized phases of the spectrum of $\mathcal{H}_L$. The two main technical challenges are the proof of a two-points or Minami estimate, as well as an estimate on the convergence to equilibrium of a hypoelliptic diffusion, the proof of which relies on Malliavin calculus and the theory of hypocoercivity.

Laure Dumaz, Cyril Labbé

• 19 November 2021

Let $H\subset G$ be semisimple Lie groups, $\Gamma\subset G$ a lattice and $K$ a compact subgroup of $G$. For $n \in \mathbb N$, let $\mathcal O_n$ be the projection to $\Gamma \backslash G/K$ of a finite union of closed $H$-orbits in $\Gamma \backslash G$. In this very general context of homogeneous dynamics, we prove an equidistribution theorem for intersections of $\mathcal O_n$ with an analytic subvariety $S$ of $G/K$ of complementary dimension: if $\mathcal O_n$ is equidistributed in $\Gamma \backslash G/K$, then the signed intersection measure of $S \cap \mathcal O_n$ normalized by the volume of $\mathcal O_n$ converges to the restriction to $S$ of some $G$-invariant closed form on $G/K$. We give general tools to determine this closed form and compute it in some examples. As our main application, we prove that, if $\mathbb V$ is a polarized variation of Hodge structure of weight $2$ and Hodge numbers $(q,p,q)$ over a base $S$ of dimension $rq$, then the (non-exceptional) locus where the Picard rank is at least $r$ is equidistributed in $S$ with respect to the volume form $c_q^r$, where $c_q$ is the $q^{\textrm{th}}$ Chern form of the Hodge bundle. This generalizes a previous work of the first author which treated the case $q=r=1$. We also prove an equidistribution theorem for certain families of CM points in Shimura varieties, and another one for Hecke translates of a divisor in $\mathcal A_g$.

Salim Tayou, Nicolas Tholozan

• 18 November 2021

It has been known since Lanford [22] that the dynamics of a hard sphere gas is described in the low density limit by the Boltzmann equation, at least for short times. The classical strategy of proof fails for longer times, even close to equilibrium. In this paper, we introduce a weak convergence method coupled with a sampling argument to prove that the covariance of the fluctuation field around equilibrium is governed by the linearized Boltzmann equation globally in time (including in diffusive regimes). This method is much more robust and simple than the one devised in [4] which was specific to the 2D case.

Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond, Sergio Simonella

• 15 November 2021

In this paper, we present a probabilistic study of rare phenomena of the cubic nonlinear Schrödinger equation on the torus in a weakly nonlinear setting. This equation has been used as a model to numerically study the formation of rogue waves in deep sea. Our results are twofold: first, we introduce a notion of criticality and prove a Large Deviations Principle (LDP) for the subcritical and critical cases. Second, we study the most likely initial conditions that lead to the formation of a rogue wave, from a theoretical and numerical point of view. Finally, we propose several open questions for future research.

Miguel Angel Garrido, Ricardo Grande, Kristin M Kurianski, Gigliola Staffilani

• 8 November 2021

This paper is dedicated to the study of a one-dimensional congestion model, consisting of two different phases. In the congested phase, the pressure is free and the dynamics is incompressible, whereas in the non-congested phase, the fluid obeys a pressureless compressible dynamics. We investigate the Cauchy problem for initial data which are small perturbations in the non-congested zone of travelling wave profiles. We prove two different results. First, we show that for arbitrarily large perturbations, the Cauchy problem is locally well-posed in weighted Sobolev spaces. The solution we obtain takes the form (vs, us)(t, x − x(t)), where x < x(t) is the congested zone and x > x(t) is the non-congested zone. The set {x = x(t)} is a free surface, whose evolution is coupled with the one of the solution. Second, we prove that if the initial perturbation is sufficiently small, then the solution is global. This stability result relies on coercivity properties of the linearized operator around a travelling wave, and on the introduction of a new unknown which satisfies better estimates than the original one. In this case, we also prove that travelling waves are asymptotically stable.

Anne-Laure Dalibard, Charlotte Perrin

• 5 November 2021

Let $F$ be a finite unramified extension of $\mathbb Q_p$ and $\bar\rho$ be an absolutely irreducible mod~$p$ $2$-dimensional representation of the absolute Galois group of $F$. Let $t$ be a tame inertial type of $F$. We conjecture that the deformation space parametrizing the potentially Barsotti--Tate liftings of $\bar\rho$ having type $t$ depends only on the Kisin variety attached to the situation, enriched with its canonical embedding into $(\mathbb P^1)^f$ and its shape stratification. We give evidences towards this conjecture by proving that the Kisin variety determines the cardinality of the set of common Serre weights $D(t,\bar\rho) = D(t) \cap D(\bar\rho)$. Besides, we prove that this dependance is nondecreasing (the smaller is the Kisin variety, the smaller is the number of common Serre weights) and compatible with products (if the Kisin variety splits as a product, so does the number of weights).

Xavier Caruso, Agnès David, Ariane Mézard

• 30 October 2021

Let $F$ be a finite unramified extension of $\mathbb Q_p$ and $\bar\rho$ be an absolutely irreducible mod~$p$ $2$-dimensional representation of the absolute Galois group of $F$. Let $t$ be a tame inertial type of $F$. We conjecture that the deformation space parametrizing the potentially Barsotti--Tate liftings of $\bar\rho$ having type $t$ depends only on the Kisin variety attached to the situation, enriched with its canonical embedding into $(\mathbb P^1)^f$ and its shape stratification. We give evidences towards this conjecture by proving that the Kisin variety determines the cardinality of the set of common Serre weights $D(t,\bar\rho) = D(t) \cap D(\bar\rho)$. Besides, we prove that this dependance is nondecreasing (the smaller is the Kisin variety, the smaller is the number of common Serre weights) and compatible with products (if the Kisin variety splits as a product, so does the number of weights).

Frédéric Jaeck, Angelo Guerraggio, Laurent Mazliak

• 30 October 2021

Let $F$ be a finite unramified extension of $\mathbb Q_p$ and $\bar\rho$ be an absolutely irreducible mod~$p$ $2$-dimensional representation of the absolute Galois group of $F$. Let $t$ be a tame inertial type of $F$. We conjecture that the deformation space parametrizing the potentially Barsotti--Tate liftings of $\bar\rho$ having type $t$ depends only on the Kisin variety attached to the situation, enriched with its canonical embedding into $(\mathbb P^1)^f$ and its shape stratification. We give evidences towards this conjecture by proving that the Kisin variety determines the cardinality of the set of common Serre weights $D(t,\bar\rho) = D(t) \cap D(\bar\rho)$. Besides, we prove that this dependance is nondecreasing (the smaller is the Kisin variety, the smaller is the number of common Serre weights) and compatible with products (if the Kisin variety splits as a product, so does the number of weights).

Frédéric Jaeck, Olivia Constantin

• 30 October 2021

As a generalization of the free semigroup algebras considered by Davidson and Pitts, and others, the second author and D.W. Kribs initiated a study of reflexive algebras associated with directed graphs. A free semigroupoid algebra L G \mathcal {L}_G is generated by a family of partial isometries, and initial projections, which act on a generalized Fock space spawned by the directed graph G G . We show that if the graph is finite, then L G \mathcal {L}_G is hyper-reflexive.

Frédéric Jaeck, Stephen Power

• 30 October 2021

As a generalization of the free semigroup algebras considered by Davidson and Pitts, and others, the second author and D.W. Kribs initiated a study of reflexive algebras associated with directed graphs. A free semigroupoid algebra L G \mathcal {L}_G is generated by a family of partial isometries, and initial projections, which act on a generalized Fock space spawned by the directed graph G G . We show that if the graph is finite, then L G \mathcal {L}_G is hyper-reflexive.

Frédéric Jaeck

• 29 October 2021

We address the problem of catching all speed $1$ geodesics of a Riemannian manifold with a moving ball: given a compact Riemannian manifold $(M,g)$ and small parameters $\e>0$ and $v>0$, is it possible to find $T>0$ and an absolutely continuous map $x:[0,T]\rightarrow M, t\mapsto x(t)$ satisfying $\|\dot{x}\|_{\infty}\leq v$ and such that any geodesic of $(M,g)$ traveled at speed $1$ meets the open ball $B_g(x(t),\e)\subset M$ within time $T$? Our main motivation comes from the control of the wave equation: our results show that the controllability of the wave equation can sometimes be improved by allowing the domain of control to move adequately, even very slowly. We first prove that, in any Riemannian manifold $(M,g)$ satisfying a geodesic recurrence condition (GRC), our problem has a positive answer for any $\varepsilon>0$ and $v>0$, and we give examples of Riemannian manifolds $(M,g)$ for which (GRC) is satisfied. Then, we build an explicit example of a domain $X\subset\R^2$ (with flat metric) containing convex obstacles, not satisfying (GRC), for which our problem has a negative answer if $\e$ and $v$ are small enough, i.e., no sufficiently small ball moving sufficiently slowly can catch all geodesics of $X$.

Cyril Letrouit

• 27 October 2021

We study the norm of the two-dimensional Brownian motion conditioned to stay outside the unit disk at all times. We obtain sharp results on the rate of escape to infinity of the process of future minima. For this, we introduce a renewal structure attached to record times and values. Additional results are given for the long time behavior of the norm.

Orphée Collin, Francis Comets

• 11 October 2021

It is well-known that observability (and, by duality, controllability) of the elliptic wave equation, i.e., with a Riemannian Laplacian, in time $T_0$ is almost equivalent to the Geometric Control Condition (GCC), which stipulates that any geodesic ray meets the control set within time $T_0$. We show that in the subelliptic setting, GCC is never verified, and that subelliptic wave equations are never observable in finite time. More precisely, given any subelliptic Laplacian $\Delta=-\sum_{i=1}^m X_i^*X_i$ on a manifold $M$, and any measurable subset $\omega\subset M$ such that $M\backslash \omega$ contains in its interior a point $q$ with $[X_i,X_j](q)\notin \text{Span}(X_1,\ldots,X_m)$ for some $1\leq i,j\leq m$, we show that for any $T_0>0$, the wave equation with subelliptic Laplacian $\Delta$ is not observable on $\omega$ in time $T_0$. The proof is based on the construction of sequences of solutions of the wave equation concentrating on geodesics (for the associated sub-Riemannian distance) spending a long time in $M\backslash \omega$. As a counterpart, we prove a positive result of observability for the wave equation in the Heisenberg group, where the observation set is a well-chosen part of the phase space.

Cyril Letrouit