Eigenfunctions and Random Waves in the Benjamini-Schramm limit

We investigate the asymptotic behavior of eigenfunctions of the Laplacian on Riemannian manifolds. We show that Benjamini-Schramm convergence provides a unified language for the level and eigenvalue aspects of the theory. As a result, we present a mathematically precise formulation of Berry's conjecture for a compact negatively curved manifold and formulate a Berry-type conjecture for sequences of locally symmetric spaces. We prove some weak versions of these conjectures. Using ergodic theory, we also analyze the connections of these conjectures to Quantum Unique Ergodicity.Comment: 40 page

Quantum Ergodicity and Averaging Operators on the Sphere

We prove quantum ergodicity for certain orthonormal bases of $L^2(\mathbb{S}^2)$, consisting of joint eigenfunctions of the Laplacian on $\mathbb{S}^2$ and the discrete averaging operator over a finite set of rotations, generating a free group. If in addition the rotations are algebraic we give a quantified version of this result. The methods used also give a new, simplified proof of quantum ergodicity for large regular graphs.Comment: 27 page

Quantum ergodicity for Eisenstein series on hyperbolic surfaces of large genus

We give a quantitative estimate for the quantum variance on hyperbolic surfaces in terms of geometric parameters such as the genus, number of cusps and injectivity radius. It implies a delocalisation result of quantum ergodicity type for eigenfunctions of the Laplacian on hyperbolic surfaces of finite area that Benjamini-Schramm converge to the hyperbolic plane. We show that this is generic for Mirzakhani's model of random surfaces chosen uniformly with respect to the Weil-Petersson volume. Depending on the particular sequence of surfaces considered this gives a result of delocalisation of most cusp forms or Eisenstein series.Comment: 21 page

L^p Norms of Eigenfunctions on Regular Graphs and on the Sphere

Abstract We prove upper bounds on the $L^p$ norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the $L^p$ norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite collection of algebraic rotations of the two-sphere. Under mild conditions, such joint eigenfunctions are shown to satisfy for large $p$ the same bounds as those known for Laplace eigenfunctions on a surface of non-positive curvature.</jats:p

Semaine d'Etude Mathématiques et Entreprises 1 : Géométrie des matrices de covariance pour le traitement de signaux radars

Les radars Doppler permettent de détecter des objets volants petits ou de faible signature radar. Thalès propose ici de réfléchir aux techniques permettant de faire ressortir de la masse des données radar celles qui sont "aberrantes" afin de repérer parmi le bruit de fond dû aux milieux environnants (nuages de pluie,...) la trace d'un objet volant

Cauliflower fractal forms arise from perturbations of floral gene networks

[EN] Throughout development, plant meristems regularly produce organs in defined spiral, opposite, or whorl patterns. Cauliflowers present an unusual organ arrangement with a multitude of spirals nested over a wide range of scales. How such a fractal, self-similar organization emerges from developmental mechanisms has remained elusive. Combining experimental analyses in an Arabidopsis thaliana cauliflower-like mutant with modeling, we found that curd self-similarity arises because the meristems fail to form flowers but keep the "memory" of their transient passage in a floral state. Additional mutations affecting meristem growth can induce the production of conical structures reminiscent of the conspicuous fractal Romanesco shape. This study reveals how fractal-like forms may emerge from the combination of key, defined perturbations of floral developmental programs and growth dynamics.This work was supported by the INRAE Caulimodel project (to F.P. and C.Go.); Inria Project Lab Morphogenetics (to C.Go., E.A., and F.P.); the ANR BBSRC Flower model project (to F.P. and C.Go.); the GRAL LabEX (ANR-10-LABX-49-01) within the framework of the CBH-EUR-GS (ANR-17-EURE-0003) (to F.P., G.T., M.L.M., and J.L.); the EU H2020 773875 ROMI project (to C.Go.); and the Spanish Ministerio de Ciencia Innovacion and FEDER (grant no. PGC2018-099232-B-I00 to F.M.).Azpeitia, E.; Tichtinsky, G.; Le Masson, M.; Serrano-Mislata, A.; Lucas, J.; Gregis, V.; Gimenez, C.... (2021). Cauliflower fractal forms arise from perturbations of floral gene networks. Science. 373(6551):1-6. https://doi.org/10.1126/science.abg5999S16373655

Autoantibodies against type I IFNs in patients with life-threatening COVID-19

Interindividual clinical variability in the course of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection is vast. We report that at least 101 of 987 patients with life-threatening coronavirus disease 2019 (COVID-19) pneumonia had neutralizing immunoglobulin G (IgG) autoantibodies (auto-Abs) against interferon-w (IFN-w) (13 patients), against the 13 types of IFN-a (36), or against both (52) at the onset of critical disease; a few also had auto-Abs against the other three type I IFNs. The auto-Abs neutralize the ability of the corresponding type I IFNs to block SARS-CoV-2 infection in vitro. These auto-Abs were not found in 663 individuals with asymptomatic or mild SARS-CoV-2 infection and were present in only 4 of 1227 healthy individuals. Patients with auto-Abs were aged 25 to 87 years and 95 of the 101 were men. A B cell autoimmune phenocopy of inborn errors of type I IFN immunity accounts for life-threatening COVID-19 pneumonia in at least 2.6% of women and 12.5% of men