La notion de représentations Anosov s'est révélée ces dernières années comme un bon analogue de celle de représentations convexe-cocompactes pour les espaces symétrique de rang supérieur. Nous nous tâcherons dans un premier temps d'expliquer comment elles sont reliées à la géométrie projective. Notre exposé s'articulera ensuite autour de l'étude de différents invariants : exposants critiques, entropies, et dimension de Hausdorff dans le cas général des sous-groupes de SL(n,R) et dans celui plus spécifique des représentations de SO(p,q). Nous présenterons enfin deux résultats de rigidités pour ces invariants. Ces travaux sont en commun avec D. Monclair et D. Monclair -- N. Tholozan.

À la croisée de la théorie de la mesure et de la géométrie, l’inégalité de Pólya–Szegő affirme que l’énergie des fonctions de Sobolev est décroissante pour le réarrangement symétrique décroissant. Cette inégalité, en lien avec l’inégalité isopérimétrique (et donc l’inégalité arithmético-géométrique), a notamment permis d’expliciter des maximiseurs dans l’inégalité de Sobolev. Je sais, ça fait beaucoup d’inégalités, mais j’introduirai toutes les notions nécessaires, et j’apporterai des preuves (élémentaires !) des résultats.

Après avoir rappelé les principales caractéristiques de la transformation de Fourier sur ${\mathbb R}^n$, nous introduirons le groupe d'Heisenberg et expliquerons de manière très élémentaire comment l'on peut voir la transformation de Fourier dans ce cadre comme fonction continue sur un espace singulier (l'espace des fréquences) que nous définirons de manière explicite.

Texte Colloquium

In this talk we discuss the change-point detection problem when dealing with complex data.

Our goal is to present a new procedure involving positive semidefinite kernels and allowing us for detecting abrupt changes arising in the full distribution of the observations along the time (and not only in their means).

The two-stage procedure we introduce involves dynamic programming and a new $l_0$-type penalty derived from a new concentration inequality applying to vectors in a reproducing kernel Hilbert space. The performance of the resulting change-point detection procedure is theoretically grounded by means of a non-asymptotic model selection result (oracle inequality).

We will also illustrate the practical behavior of our kernel change-point procedure on a wide range of simulated data. In particular we empirically validate our penalty since the resulting penalized criterion recovers the true (number of) change-points with high probability.

We will finally discuss the influence of the kernel on the results in practice.

In this talk we consider the estimation and inference problem of interval censored data. These types of data arise when patients are followed-up at different visits and the exact occurence of the event of interest is unknown. Instead, one only knows that the event has occurred between two time visits. These data also encompass left-censored observations (when the event has occurred before the first visit) and right-censored data (when the event has not yet occurred after the last follow-up time). We study the nonparametric and regression settings by specifying a piecewise constant function for the hazard rate. Treating the true event times of interest as unobserved data, the EM algorithm is implemented. In order to determine the number and locations of the cuts of the hazard function, a L0 penalized likelihood method is used, such that a large grid of cuts is initially implemented and the penalization technique forces two similar adjacent values to be equal. Statistical inference of the model parameters are derived from likelihood theory. The method is illustrated on a dental dataset where 322 patients with 400 avulsed and replanted permanent teeth were followed-up prospectively in the period from 1965 to 1988 at the university hospital in Copenhagen, Denmark. The following replantation procedure was used: the avulsed tooth was placed in saline as soon as the patient was received at the emergency ward. If the tooth was obviously contaminated, it was cleansed with gauze soaked in saline or rinsed with a flow of saline from a syringe. The tooth was replanted in its socket by digital pressure. The patients were then examined at regular visits to the dentist. In this study, we focused on a complication called ankylosis such that the variable of interest is the time from replantation of the tooth to ankylosis. 28% of the data were left censored, 35.75% were interval censored and 36.25% were right censored. A Cox model was implemented on this dataset and showed that the stage of root formation (mature or immature tooth) and the length of extra-alveolar storage time were significantly associated with the risk of experiencing ankylosis.

This is a joint work with Grégory Nuel (DR CNRS, LPSM, Paris 6) and Eva Lauridsen (Department of Pediatric Dentistry and Clinical Genetics, School of Dentistry, University of Copenhagen).

In the first part of the talk, we will introduce spatial Gaussian processes. Spatial Gaussian processes are widely studied from a statistical point of view, and have found applications in many fields, including geostatistics, climate science and computer experiments. Exact inference can be conducted for Gaussian processes, thanks to the Gaussian conditioning theorem. Furthermore, covariance parameters can be estimated, for instance by Maximum Likelihood.In the second part of the talk, we will introduce a class of iterative sampling strategies for Gaussian processes, called 'stepwise uncertainty reduction' (SUR). We will give examples of SUR strategies which are widely applied to computer experiments, for instance for optimization or detection of failure domains. We will provide a general consistency result for SUR strategies, together with applications to the most standard examples.