Publications

2020

• Quasi-isometric invariance of continuous group $L^p$-cohomology, and first applications to vanishings
  
  • Bourdon Marc
  • Remy Bertrand
  , 2020. We show that the continuous L p-cohomology of locally compact second countable groups is a quasi-isometric invariant. As an application, we prove partial results supporting a positive answer to a question asked by M. Gromov, suggesting a classical behaviour of continuous L p-cohomology of simple real Lie groups. In addition to quasi-isometric invariance, the ingredients are a spectral sequence argument and Pansu's vanishing results for real hyperbolic spaces. In the best adapted cases of simple Lie groups, we obtain nearly half of the relevant vanishings. (10.5802/ahl.61)
  DOI : 10.5802/ahl.61
• On the Vlasov-Maxwell System : régularity and non-relativistic limit
  
  • Brigouleix Nicolas
  , 2020. In this dissertation, we study the Vlasov-Maxwell system of partial differential equations, describing the evolution of the distribution function of charged particles in a plasma. More precisely, we study the regularity of solutions to this system, and the question of the non-relativstic limit.In the first part, we study a Toy-model, combining the Vlasov equation with a system of transport equations. We use the methods developed to obtain and imrpove the Glassey-Strauss criterion, which gives a sufficient condition under which strong solutions do not develop singularities. The loss of regularity occures when the speed of the particles is close to the characteristic speed of the joined hyperbolic system. The same phenomenon occures for the solutions of the Toy-model, but its structure is easier to handle.In the second part, we focus on the question of the non-relativistic limit. After a rescaling of the equations, the speed of light can be considered as a big parameter. When it tends to infinity, it is called the non-relativistic limit. At first order, the non-relativistic limit of the Vlasov-Maxwell system is the Vlasov-Poisson system. First, an iterative method giving arbitrary high non-relativistic approximations is established. These systems combine the Vlasov-equation with elliptic systems of equations, and are well-posed in some weigthed Sobolev spaces. We also prove a result on the non-relativistic limit to the Vlasov-Poisson system under the weaker assumption of boundedness of the macroscopic density. We study a functional quantifying the Wasserstein distance between weak solutions of both systems.
• Epidemic Dynamics via Wavelet Theory and Machine Learning with Applications to Covid-19
  
  • Tat Dat Tô
  • Protin Frédéric
  • Hang Nguyen T. T.
  • Jules Martel
  • Duc Thang Nguyen
  • Piffault Charles
  • Willy Rodríguez
  • Susely Figueroa
  • Lê Hông Vân
  • Tuschmann Wilderich
  • Tien Zung Nguyen
  Biology, MDPI, 2020, 9 (12), pp.477. We introduce the concept of epidemic-fitted wavelets which comprise, in particular, as special cases the number I(t) of infectious individuals at time t in classical SIR models and their derivatives. We present a novel method for modelling epidemic dynamics by a model selection method using wavelet theory and, for its applications, machine learning-based curve fitting techniques. Our universal models are functions that are finite linear combinations of epidemic-fitted wavelets. We apply our method by modelling and forecasting, based on the Johns Hopkins University dataset, the spread of the current Covid-19 (SARS-CoV-2) epidemic in France, Germany, Italy and the Czech Republic, as well as in the US federal states New York and Florida. (10.3390/biology9120477)
  DOI : 10.3390/biology9120477
• Approximation of non-archimedean Lyapunov exponents and applications over global fields
  
  • Gauthier Thomas
  • Okuyama Yûsuke
  • Vigny Gabriel
  Transactions of the American Mathematical Society, American Mathematical Society, 2020, 373 (12), pp.8963-9011. (10.1090/tran/8232)
  DOI : 10.1090/tran/8232
• Quasi-isometric invariance of continuous group L p -cohomology, and first applications to vanishings
  
  • Bourdon Marc
  • Rémy Bertrand
  Annales Henri Lebesgue, UFR de Mathématiques - IRMAR, 2020, 3, pp.1291-1326. We show that the continuous L p-cohomology of locally compact second countable groups is a quasi-isometric invariant. As an application, we prove partial results supporting a positive answer to a question asked by M. Gromov, suggesting a classical behaviour of continuous L p-cohomology of simple real Lie groups. In addition to quasi-isometric invariance, the ingredients are a spectral sequence argument and Pansu's vanishing results for real hyperbolic spaces. In the best adapted cases of simple Lie groups, we obtain nearly half of the relevant vanishings. (10.5802/ahl.61)
  DOI : 10.5802/ahl.61
• Large time behavior of small data solutions to the Vlasov-Navier-Stokes system on the whole space
  
  • Han-Kwan Daniel
  , 2020. We study the large time behavior of small data solutions to the Vlasov-Navier-Stokes system on $\R^3 \times \R^3$. We prove that the kinetic distribution function concentrates in velocity to a Dirac mass supported at $0$, while the fluid velocity homogenizes to $0$, both at a polynomial rate. The proof is based on two steps, following the general strategy laid out in \cite{HKMM}: (1) the energy of the system decays with polynomial rate, assuming a uniform control of the kinetic density, (2) a bootstrap argument allows to obtain such a control. This last step requires a fine understanding of the structure of the so-called Brinkman force, which follows from a family of new identities for the dissipation (and higher versions of it) associated to the Vlasov-Navier-Stokes system.
• Relative trace formula for compact quotient and pseudocoefficients for relative discrete series
  
  • Delorme Patrick
  • Harinck Pascale
  International Mathematics Research Notices, Oxford University Press (OUP), 2020. We introduce the notion of relative pseudocoefficient for relative discrete series of real spherical homogeneous spaces of reductive groups. We prove that such relative pseudocoefficient does not exist for semisimple symmetric spaces of type G(C)/G(R) and construct strong relative pseudocoefficients for some hyperbolic spaces. We establish a toy model for the relative trace formula of H.Jacquet for compact discrete quotient {\Gamma}\G. This allows us to prove that a relative discrete series which admits strong pseudocoefficient with sufficiently small support occurs in the spectral decomposition of L^2({\Gamma}\G) with a nonzero period. (10.1093/imrn/rnaa248)
  DOI : 10.1093/imrn/rnaa248
• Positivity of direct images and projective varieties with nonnegative curvature
  
  • Wang Juanyong
  , 2020. The birational classification of algebraic varieties is a central problem in algebraic geometry. Recently great progress has been made towards the establishment of the MMP and the Abundance and by these works, smooth (or mildly singular) projective varieties can be birationally divided into two categories: 1. varieties with pseudoeffective canonical divisor, which are shown to reach a minimal model under the MMP; 2. uniruled varieties, which are covered by rational curves. In this thesis refined studies of these two categories of varieties are carried out respectively, by following the philosophy of studying the canonical fibrations associated to them.For any variety X in the first category, the most important canonical fibration associated to X is the Iitaka-Kodaira fibration whose base variety is of dimension equal to the Kodaira dimension of X. This thesis tacles an important corollary of the Abundance conjecture, namely, the Iitaka conjecture C_{n,m}, which states the supadditivity of the Kodaira dimension with respect to algebraic fibre spaces. In this thesis the Kähler version of C_{n,m} is proved under the assumption that the base variety of the fibre space is a complex torus by further developping the positivity theorem of direct images and the pluricanonical version of the Green-Lazarsfeld-Simpson type theorem on cohomology jumping loci. This generalizes the main result of Cao-Păun (2017).As for varieties in the second category, one studies the Albanese map and the MRC fibration, instead of the Iitaka-Kodaira fibration. A philosophy in this investigation is that when the tangent bundle or the anticanonical divisor admits certain positivity, the aforementioned two fibrations of the variety should have a rigid structure. In this thesis I study in this thesis the structure of (mildly singular) projective varieties with nef anticanonical divisor. By again applying the positivity of direct images and by applying results from the foliation theory, I manage to prove that the Albanese map of such variety is a locally constant fibration and that if its smooth locus is simply connected then the MRC fibration induces a splitting into a product. These generalize the corresponding results for smooth projective varieties in Cao (2019) and Cao-Höring (2019)
• Relative regular Riemann-Hilbert correspondence
  
  • Fiorot Luisa
  • Monteiro Fernandes Teresa
  • Sabbah Claude
  Proceedings of the London Mathematical Society, London Mathematical Society, 2020, 123 (6), pp.649-654. (10.1112/plms.12362)
  DOI : 10.1112/plms.12362
• Construction of multi-bubble solutions for the energy-critical wave equation in dimension 5
  
  • Jendrej Jacek
  • Martel Yvan
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2020, 139, pp.317-355. (10.1016/j.matpur.2020.02.007)
  DOI : 10.1016/j.matpur.2020.02.007
• Quantum optimal transport is cheaper
  
  • Caglioti Emanuele
  • Golse François
  • Paul Thierry
  Journal of Statistical Physics, Springer Verlag, 2020, 181 ((1),), pp.149-162. We compare bipartite (Euclidean) matching problems in classical and quantum mechanics. The quantum case is treated in terms of a quantum version of the Wasserstein distance introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016), 165-205]. We show that the optimal quantum cost can be cheaper than the classical one. We treat in detail the case of two particles: the equal mass case leads to equal quantum and classical costs. Moreover, we show examples with different masses for which the quantum cost is strictly cheaper than the classical cost.
• Asymptotic expansion of the mean-field approximation
  
  • Paul Thierry
  • Pulvirenti Mario
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2020, 39 ((4)), pp.1891-1921. We established and estimate the full asymptotic expansion in integer powers of 1 N of the [ √ N ] first marginals of N-body evolutions lying in a general paradigm containing Kac models and non-relativistic quantum evolution. We prove that the coefficients of the expansion are, at any time, explicitly computable given the knowledge of the linearization on the one-body meanfield kinetic limit equation. Instead of working directly with the corresponding BBGKY-type hierarchy, we follows a method developed in [22] for the meanfield limit, dealing with error terms analogue to the v-functions used in previous works. As a by-product we get that the rate of convergence to the meanfield limit in 1 N is optimal.
• Fractional Fokker–Planck Equation with General Confinement Force
  
  • Lafleche Laurent
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2020, 52 (1), pp.164-196. This article studies a Fokker-Planck type equation of fractional diffusion with conservative drift ∂f/∂t = ∆^(α/2) f + div(Ef), where ∆^(α/2) denotes the fractional Laplacian and E is a confining force field. The main interest of the present paper is that it applies to a wide variety of force fields, with a few local regularity and a polynomial growth at infinity. We first prove the existence and uniqueness of a solution in weighted Lebesgue spaces depending on E under the form of a strongly continuous semigroup. We also prove the existence and uniqueness of a stationary state, by using an appropriate splitting of the fractional Laplacian and by proving a weak and strong maximum principle. We then study the rate of convergence to equilibrium of the solution. The semigroup has a property of regularization in fractional Sobolev spaces, as well as a gain of integrability and positivity which we use to obtain polynomial or exponential convergence to equilibrium in weighted Lebesgue spaces. (10.1137/18M1188331)
  DOI : 10.1137/18M1188331
• Mathematics and music: loves and fights
  
  • Paul Thierry
  , 2020. We present different aspects of the special relationship that music has with mathematics, in particular the concepts of rigour and realism in both fields. These directions are illustrated by comments on the personal relationship of the author with Jean-Claude, together with examples taken from his own works, specially the "Duos pour un pianiste".
• Stability for small data: the drift model of the conformal method
  
  • Vâlcu Caterina
  Classical and Quantum Gravity, IOP Publishing, 2020, 37 (19), pp.195028. The conformal method in general relativity aims at successfully parametrising the set of all initial data associated with globally hyperbolic spacetimes. One such mapping was suggested by Maxwell D (2014 Initial data in general relativity described by expansion, conformal deformation and drift (arXiv:1407.1467)). For closed manifolds, I verify that the solutions of the corresponding conformal system are stable, in the sense that they present a priori bounds under perturbations of the system’s coefficients. This result holds in dimensions 3 ⩽ n ⩽ 5, when the metric is conformally flat, the drift is small. A scalar field with suitably high potential is considered in this case. (10.1088/1361-6382/abadb0)
  DOI : 10.1088/1361-6382/abadb0
• Solutions blowing up on any given compact set for the energy subcritical nonlinear wave equation
  
  • Cazenave Thierry
  • Martel Yvan
  • Zhao Lifeng
  Journal of Differential Equations, Elsevier, 2020. (10.1016/j.jde.2019.08.030)
  DOI : 10.1016/j.jde.2019.08.030
• Topological computation of some Stokes phenomena on the affine line
  
  • D’agnolo Andrea
  • Hien Marco
  • Morando Giovanni
  • Sabbah Claude
  Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2020, 70 (2), pp.739-808. (10.5802/aif.3323)
  DOI : 10.5802/aif.3323
• THE ARITHMETIC OF POLYNOMIAL DYNAMICAL PAIRS
  
  • Favre Charles
  • Gauthier Thomas
  , 2020. We study one-dimensional algebraic families of pairs given by a polynomial with a marked point. We prove an "unlikely intersection" statement for such pairs thereby exhibiting strong rigidity features for these pairs. We infer from this result the dynamical André-Oort conjecture for curves in the moduli space of polynomials, by describing one-dimensional families in this parameter space containing infinitely many post-critically finite parameters.
• Large time behavior of the Vlasov-Navier-Stokes system on the torus
  
  • Han-Kwan Daniel
  • Moussa Ayman
  • Moyano Iván
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2020, 236 (3), pp.1273-1323. We study the large time behavior of Fujita–Kato type solutions to the Vlasov–Navier–Stokes system set on $\T^3 \times \R^3$. Under the assumption that the initial so-called modulated energy is small enough, we prove that the distribution function converges to a Dirac mass in velocity, with exponential rate. The proof is based on the fine structure of the system and on a bootstrap analysis allowing us to get global bounds on moments. (10.1007/s00205-020-01491-w)
  DOI : 10.1007/s00205-020-01491-w
• Uniqueness of the solution to the 2D Vlasov-Navier-Stokes system
  
  • Han-Kwan Daniel
  • Miot Evelyne
  • Moussa Ayman
  • Moyano Iván
  Revista Matemática Iberoamericana, European Mathematical Society, 2020, 1 (36), pp.37-60. We prove a uniqueness result for weak solutions to the Vlasov-Navier-Stokes system in two dimensions, both in the whole space and in the periodic case, under a mild decay condition on the initial distribution function. The main result is achieved by combining methods from optimal transportation (introduced in this context by G. Loeper) with the use of Hardy's maximal function, in order to obtain some fine Wassestein-like estimates for the difference of two solutions of the Vlasov equation. (10.4171/RMI/1120)
  DOI : 10.4171/RMI/1120
• Local Zeta functions for a class of p-adic symmetric spaces (I)
  
  • Harinck Pascale
  • Rubenthaler Hubert
  , 2020. {\bf Abstract:}\,{\it This is an extended version of the first part of a forthcoming paper where we will study the local Zeta functions of the minimal spherical series for the symmetric spaces arising as open orbits of the parabolic prehomogeneous spaces of commutative type over a p-adic field. The case where the ground field is $\R$ has already been considered by Nicole Bopp and the second author (\cite{BR}). If $F$ is a p-adic field of caracteristic $0$, we consider a reductive Lie algebra $\widetilde{\go g}$ over $F$ which is endowed with a short $\Z$-grading: $\widetilde{\go g}=\go{g}_{-1}\oplus \go{g}_{0}\oplus \go{g}_{1}$. We also suppose that the representation $(\go{g}_{0}, \go{g}_{1})$ is absolutely irreducible. Under a so-called regularity condition we study the orbits of $G_{0}$ in $\go{g}_{1}$, where $G_{0}$ is an algebraic group defined over $F$, whose Lie algebra is $\go{g}_{0}$. We also investigate the $P$-orbits, where $P$ is a minimal $\sigma$-split parabolic subgroup of $G$ ($\sigma$ being the involution which defines a structure of symmetric space on any open $G_{0}$-orbit in $\go{g}_{1}$).}
• THE RANDOM BATCH METHOD FOR N -BODY QUANTUM DYNAMICS
  
  • Golse François
  • Jin Shi
  • Paul Thierry
  Journal of Computational Mathematics -International Edition-, Global Science Press, 2020. This paper discusses a numerical method for computing the evolution of large interacting system of quantum particles. The idea of the random batch method is to replace the total interaction of each particle with the N-1 other particles by the interaction with p<N particles chosen at random at each time step, multiplied by (N-1)/p. This reduces the computational cost of computing the interaction partial per time step from O(N^2) to O(N). For simplicity, we consider only in this work the case p=1 --- in other words, we assume that N is even, and that at each time step, the N particles are organized in N/2 pairs, with a random reshuffling of the pairs at the beginning of each time step. We obtain a convergence estimate for the Wigner transform of the single-particle reduced density matrix of the particle system at time t that is uniform in N>1 and independent of the Planck constant.
• On the Convergence of Time Splitting Methods for Quantum Dynamics in the Semiclassical Regime
  
  • Golse François
  • Jin Shi
  • Paul Thierry
  Foundations of Computational Mathematics, Springer Verlag, 2020, pp.DOI: 10.1007/s10208-020-09470-z.. By using the pseudo-metric introduced in [F. Golse, T. Paul: Archive for Rational Mech. Anal. 223 (2017) 57-94], which is an analogue of the Wasserstein distance of exponent 2 between a quantum density operator and a classical (phase-space) density, we prove that the convergence of time splitting algorithms for the von Neumann equation of quantum dynamics is uniform in the Planck constant $\hbar$. We obtain explicit uniform in $\hbar$ error estimates for the first order Lie-Trotter, and the second order Strang splitting methods.