Exceptionnellement, l'accueil de la Cellule Informatique au bureau 225

• ne sera ouvert qu'à partir de 10h
• et sera fermé vendredi
• et il n'y aura pas d'accueil au bureau 270.

en raison de la participation d'une partie de l'équipe informatique aux Journées Mathrice qui auront lieu au laboratoire XLIM de Limoges.

• Pensez à anticiper les retraits de matériel de prêt.

Pour l'analyse des séries temporelles ou pour l'étude de la convergence de certains algorithmes stochastiques tels que ceux basés sur la dynamique de Langevin, la dynamique de certains processus markoviens à temps discret est naturellement pertubée par l'introduction d'un désordre stochastique que l'on peut appeler bruit, facteur exogène ou environnement aléatoire suivant la terminologie utilisée dans les différents domaines. L'objectif de cet exposé est de montrer que l'on peut étudier la stabilité de ces chaînes de Markov en environnement aléatoire sur des espaces non bornés à partir de versions adaptées de conditions de stabilité plus classiques (condition de drift, de petit ensemble ou contraction d'applications aléatoires). Des conditions suffisantes pour l'existence de mesures stationnaires et ergodiques ainsi qu'un contrôle des coefficients de alpha-mélange pour ces processus peuvent être obtenus.  On peut alors en deduire une théorie limite pour des processus autorégressifs avec covariables exogènes ainsi que des estimés qualitatifs de la vitesse de convergence de certains algorithmes stochastiques.

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. 

It is well understood that positivity or negativity properties of canonical line bundle encode a significant amount of geometric data about the underlying projective variety. It is therefore unsurprising to expect that the same should be true for the relative canonical divisor of families of projective varieties. For families of varieties whose canonical divisor is ample (canonically polarized) or numerically trivial (Calabi-Yau), important positivity properties of the pushforward of the relative (pluri)canonical was discovered by Fujita, Kawamata, Kollár and Viehweg. Many fundamental results then followed as a consequence - from moduli theory of such varieties to birational geometry of base spaces of their degeneration. For families of Fano varieties however much less is known. In this talk I will discuss how one can complement some of these classical results in the Fano case. This is based on ongoing joint work with Sándor Kovács.

The Calabi problem asks which compact complex manifolds are Kähler-Einstein (KE)- i.e. can be endowed with a canonical metric that satisfies both an algebraic condition (being Kähler) and the Einstein (partial differential) equation. Such manifolds always have a canonical class of definite sign. The existence of such a metric on manifolds with positive and trivial canonical class (general type and Calabi Yau) was proved in the 70s. In the case of Fano manifolds, the situation is more subtle - Fano manifolds may or may not have a KE metric. 

We now know that a Fano manifold admits a KE metric precisely when it satisfies a sophisticated algebro-geometric condition called K-polystability. Surprisingly, K-polystability also sheds some light on another important problem: while Fano varieties do not behave well in families, K-polystable Fano varieties do and form K-moduli spaces. 

Our explicit understanding of K-polystability is still partial, and few examples of K-moduli spaces are known. In dimension 3, Fano manifolds were classified into 105 deformation families by Mori-Mukai and Iskovskikh. 

 In this talk, I will present an overview of the Calabi problem and discuss its solution in dimension 3. Knowing - as we do - which families in the classification of Fano 3-folds have K-polystable members is a starting point to investigate the corresponding K-moduli spaces. I will describe explicitly some K-moduli spaces of Fano 3-folds of small dimension.