19 novembre 2020

Elisabeth Gassiat (LMO, Université Paris Saclay)
Déconvolution lorsque rien n’est connu sur le bruit.

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Résumé : In the deconvolution problem, observations come from a signal additively corrupted with independent noise.
Estimators based on Fourier transforms are the most widespread in this setting as convolution with a known error distribution translates into a multiplication of the Fourier transform of the signal by the Fourier transform of the noise. However, this assumption may have a significant impact on the robustness of deconvolution estimators as pointed out by (Meister 2004) where the author established that the mean integrated squared error of such an estimator can grow to infinity when the noise distribution is misspecified.
The subject of my talk will be to solve the deconvolution problem without any assumption on the noise distribution and based solely on a sample of observations. I will prove this is possible as soon as the signal (i) has a distribution with light enough tails and (ii) has at least two dimensions and may be decomposed into two subsets of random variables which satisfy some weak dependency assumption. This identifiability result applies to several popular statistical models in which the noise assumptions may thus be avoided.
In a joint work with S. Le Corff and L. Lehéricy, we propose an estimator of the density of the distribution of the signal which is shown to be minimax adaptive for the mean integrated squared error, with a rate depending both on the regularity and the tail lightness of the distribution of the signal.

Déconvolution lorsque rien n’est connu sur le bruit.  Version PDF

Nikita Simonov (CEREMADE-Université Paris-Dauphine)
Séminaire AN-EDP-Stability in Gagliardo-Nirenberg inequalities

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Lieu : visioconférence (lien dans l'annonce par mail)

Résumé : In some functional inequalities best constants and minimizers are known. The next question is stability : suppose that a function « almost attains the equality », in which sense it is close to one of the minimizers ? In this talk a will address a recent result on quantitative stability of a subfamily of Gagliardo-Nirengerg inequalities. The results are based on a joint work with M. Bonforte, J. Dolbeault and B. Nazaret.

Séminaire AN-EDP-Stability in Gagliardo-Nirenberg inequalities  Version PDF

Cyril Houdayer (IMO)
Théorie ergodique noncommutative des réseaux de rang supérieur

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Lieu : https://bbb.imo.universite-paris-saclay.fr/b/ram-jaz-4ow-zur

Résumé : Je vais présenter des travaux récents concernant la structure des C*-algèbres (trace, idéaux maximaux) associées aux représentations unitaires des réseaux des groupes de Lie semisimples de rank supérieur. Ces résultats étendent les travaux célèbres de Margulis (Théorème du sous-groupe normal, 1978) et Stuck-Zimmer (Rigidité des stabilisateurs, 1992). J’expliquerai le théorème principal qui fournit un critère dynamique pour l’application de Furstenberg associée aux actions stationnaires des réseaux sur des C*-algèbres (noncommutatives). Ce critère dynamique permet par ailleurs de résoudre le problème de Glasner-Weiss (2014) concernant les URS des réseaux.
Travaux en collaboration avec R. Boutonnet (arXiv:1908.07812) et U. Bader, R. Boutonnet, J. Peterson (arXiv:2009.09952).

Théorie ergodique noncommutative des réseaux de rang supérieur  Version PDF

Laura Monk (IRMA (Strasbourg))
Geometry and spectrum of random hyperbolic surfaces

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Résumé : The main aim of this talk is to present geometric and spectral properties of typical hyperbolic surfaces. More precisely, I will :

  • introduce a probabilistic model, first studied by Mirzakhani, which is a natural and convenient way to sample random hyperbolic surfaces ;
  • describe the geometric properties of these random surfaces : diameter, injectivity radius, Cheeger constant, Benjamini-Schramm convergence...
  • explain how one can deduce from this geometric information estimates of the number of eigenvalues of the Laplacian in an interval [a,b], using the Selberg trace formula.

Geometry and spectrum of random hyperbolic surfaces  Version PDF