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Déterminer l’image de l’ensemble des points rationnels sous un morphisme de variétés est une question naturelle et difficile. D’après la théorie des orbifoldes de Campana, sur un corps de nombres, cette image est contenue dans l’ensemble des points de Campana associés à la base orbifolde du morphisme. Cette première approximation est ensuite affinée par la théorie des firmaments développée par Abramovich, qui conduit à la notion de points firms. Dans ce projet en collaboration avec L.Herr, M.Pieropan et T. Poiret, nous abordons le même problème et reformulons cette notion de point firm dans le langage de la géométrie logarithmique. Cette approche nous permet de démontrer une affirmation d’Abramovich sur le relèvement des points firms le long de morphismes toroïdaux.
On s'intéresse au nombre de valeurs propres négatives de l'opérateur de Schrödinger fractionnaire $H_s=(-\Delta)^s-V(x)$ dans $L^2(\mathbb{R}^d)$, en dimension quelconque $d\ge1$, et pour tout $s\gt 0$. La littérature concernant l'opérateur de Schrödinger non fractionnaire ($s=1$) est très vaste. On rappellera en particulier les célèbres estimations de Cwikel, Lieb et Rozenblum (CLR) en dimension $d\ge3$, l'estimation de Bargmann en dimension $d=1$, et des résultats existant dans le cas critique de la dimension $2$. En dimension quelconque, une borne dans le cas sous-critique $0\lt s \lt d/2$ s'obtient de la même façon que les estimations CLR. Dans cet exposé, on s'intéressera au cas sur-critique $s\ge d/2$, incluant à la fois le cas critique $s=d/2$ et le cas de l'opérateur de Schrödinger polyharmonique où $s$ est un entier positif quelconque.
We introduce a new family of rank metric codes. The construction of these codes relies on the "Chinese Remainder Theorem" for linearised polynomials. Linearised polynomials are polynomials in which the exponents of all the monomials are powers of q and the coefficients come from an extension field of the finite field of order q. The set of these polynomials forms a right Euclidean ring. We present the lifting of the isomorphism of the Chinese remainder theorem for linearised polynomials. We show how this lifting leads to a decoding algorithm for a special case of this family of codes: the case where the linearised polynomials have coefficients in the finite field of order q. (Joint work with Olivier Ruatta and Philippe Gaborit).
A guiding problem in algebraic geometry is the classification of varieties. In dimension 1, the main invariant for their classification is the genus. Similarly, in higher dimension we study positivity properties of the canonical divisor and a first measure of these is its Iitaka dimension. A long-standing problem is how we can relate Iitaka dimensions in fibrations: the Iitaka conjecture. Recently, Chang proved an inequality for the Iitaka dimensions of the anticanonical divisors in fibrations over fields of characteristic 0. Both the Iitaka’s conjecture and Chang’s theorem are known to fail in positive characteristic. However, in a joint work with Brivio and Chang, we prove that anti-Iitaka holds when the “arithmetic properties” of the anticanonical divisor are sufficiently good.
Pas de séminaire
Le résumé de l'exposé d'Éric
We derive an entropy stable extension of the Navier-Stokes-Fourier equations into the transition regime of rarefied gases. We do this through a variational multiscale reformulation of the closure of conservation equations derived from the Boltzmann equation. Our reformulation subsumes existing methods such as the Chapman-Enskog expansion. We apply the linearized version of this extension to the stationary heat problem and the Poiseuille channel and compare our analytical solutions to asymptotic and numerical solutions of the linearized Boltzmann equation. In both model problems, our solutions compare remarkably well in the transition regime. For some macroscopic variables, this agreement even extends far beyond the transition regime.
In this talk we will consider the approximation numbers of differences of composition operators acting on the Hardy-Hilbert space $H^2(\mathbb{D})$. The component structure of bounded composition operators is a widely studied area and in order to understand whether two composition operators belong to the same component, it is important to understand how their difference behaves (compact, bounded etc.). One of the key elements in understanding the behavior of an operator is to consider its approximation numbers since it gives us the information about how much our operator differs from a bounded/compact one. During the talk we will mention how we can combine these two topics in operator theory and how one can obtain optimal upper and lower bounds for approximation numbers of differences using classical invariants like Bernstein and Gelfand numbers and specific choices of Blaschke products from the underlying function space.
Joint work with Frédéric Bayart of Laboratoire de Mathématiques Blaise Pascal.
References
[1] G. Lechner, D. Li, H. Queff ́elec, L. Rodriguez-Piazza : Approximation numbers of weighted composition operators. Journal of Functional Analysis 274, 1928–1958 (2018).
[2] J. Moorhouse, C. Toews : Differences of composition operators. Contemporary Mathematics 321, 207–213 (2003).
[3] H. Queff ́elec, K. Seip : Decay rates for approximation numbers of composition operators. Journal d’Analyse Math ́ematique 125, 371–399 (2015).
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Le résumé de l'exposé de Gilles
TBA
A préciser
The aim of this work is to present new approaches to define Wasserstein-like barycenters for Gaussian distributions and Gaussian mixtures, while imposing the marginals of the barycenter. For instance, Wasserstein barycenters do not preserve marginals in general. In this work, we first characterize sufficient and necessary conditions for the Wasserstein barycenter between two Gaussian distributions to preserve marginals, and provide necessary conditions in the case of more than two Gaussians. This preliminary analysis enable us to propose modified Wasserstein barycenters that have prescribed marginals of the distributions, both for Gaussian distributions and for mixtures of Gaussian distributions. In the case of Gaussian distributions, the marginal-constrained modified Wasserstein barycenters can be analytically computed, while for Gaussian mixtures, computing the marginal-preserving barycenter consists in a postprocessing of the Gaussian mixture Wasserstein barycenter. In both cases, we provide numerical simulations illustrating the difference between Wasserstein barycenters and modified marginal-constrained Wasserstein barycenters. We illustrate the interest of the latter for interpolation tasks between probability measures. In particular, we motivate this work by applications in quantum chemistry, for electronic structure calculations in molecules.
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Voir programme :
https://yanntraonmilin.perso.math.cnrs.fr/?p=559
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Le résumé de l'exposé d'Élise
Pas de séminaire cette semaine puisqu'il y a la conférence pour les 60 ans de Yuri Bilu : https://yubi60.pages.math.cnrs.fr/
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A définir
A définir
Institut de Mathématiques de Bordeaux UMR 5251
Université de Bordeaux
351, cours de la Libération - F 33 405 TALENCE
Tél: +33 (0)5 40 00 60 70
institut
math.u-bordeaux.fr
