## 5 événements

• Séminaire de Statistique

### Mardi 22 janvier 09:15-10:45 - Marc Hallin (Séminaire commun proba-stats) - ECARES et Département de Mathématique Université libre de Bruxelles

Center-Outward Distribution Functions, Quantiles, Ranks, and Signs in R^d : A Measure Transportation Approach

Résumé : Unlike the real line, the $d$-dimensional space $R^d$, for $d \geq 2$, is not canonically ordered. As a consequence, such fundamental and strongly order-related univariate concepts as quantile and distribution functions, and their empirical counterparts, involving ranks and signs, do not canonically extend to the multivariate context. Palliating that lack of a canonical ordering has remained an open problem for more than half a century, and has generated an abundant literature, motivating, among others, the development of statistical depth and copula-based methods. We show that, unlike the many definitions that have been proposed in the literature, the measure transportation-based ones introduced in Chernozhukov, Galichon, Hallin and Henry (2017) enjoy all the properties (distribution-freeness and the maximal invariance property that entails preservation of semiparametric efficiency) that make univariate quantiles and ranks successful tools for semiparametric statistical inference. We therefore propose a new \it center-outward definition of multivariate distribution and quantile functions, along with their empirical counterparts, for which we establish a Glivenko-Cantelli result---the quintessential property of all distribution functions. Our approach, based on results by McCann (1995), is geometric rather than analytical and, contrary to the Monge-Kantorovich one in Chernozhukov et al. (2017) (which assumes compact supports, hence finite moments of all orders), does not require any moment assumptions. The resulting ranks and signs are shown to be strictly distribution-free, and maximal invariant under the action of a data-driven class of (order-preserving) transformations generating the family of absolutely continuous distributions ; that maximal invariance, in view of a general result by Hallin and Werker (2003), is the theoretical foundation of the semiparametric efficiency preservation property of ranks. The corresponding quantiles are equivariant under the same transformations.

Lieu : Amphi Schwartz

• Séminaire de Probabilités

### Mardi 22 janvier 09:15-10:45 - Marc Hallin (Séminaire commun proba-stats) - ECARES et Département de Mathématique Université libre de Bruxelles

Center-Outward Distribution Functions, Quantiles, Ranks, and Signs in $R^d$ : A Measure Transportation Approach

Résumé : Unlike the real line, the $d$-dimensional space $R^d$, for $d \geq 2$, is not canonically ordered. As a consequence, such fundamental and strongly order-related univariate concepts as quantile and distribution functions, and their empirical counterparts, involving ranks and signs, do not canonically extend to the multivariate context. Palliating that lack of a canonical ordering has remained an open problem for more than half a century, and has generated an abundant literature, motivating, among others, the development of statistical depth and copula-based methods. We show that, unlike the many definitions that have been proposed in the literature, the measure transportation-based ones introduced in Chernozhukov, Galichon, Hallin and Henry (2017) enjoy all the properties (distribution-freeness and the maximal invariance property that entails preservation of semiparametric efficiency) that make univariate quantiles and ranks successful tools for semiparametric statistical inference. We therefore propose a new center-outward definition of multivariate distribution and quantile functions, along with their empirical counterparts, for which we establish a Glivenko-Cantelli result---the quintessential property of all distribution functions. Our approach, based on results by McCann (1995), is geometric rather than analytical and, contrary to the Monge-Kantorovich one in Chernozhukov et al. (2017) (which assumes compact supports, hence finite moments of all orders), does not require any moment assumptions. The resulting ranks and signs are shown to be strictly distribution-free, and maximal invariant under the action of a data-driven class of (order-preserving) transformations generating the family of absolutely continuous distributions ; that maximal invariance, in view of a general result by Hallin and Werker (2003), is the theoretical foundation of the semiparametric efficiency preservation property of ranks. The corresponding quantiles are equivariant under the same transformations.

Lieu : Amphithéâtre Schwartz

• Séminaire Modélisation, Analyse et Calcul

### Mardi 22 janvier 11:00-12:00 - Nuutti Hyvönen - Aalto University, Espoo

Electrical impedance tomography under incomplete information about the measurement setup

Résumé : The aim of electrical impedance tomography (EIT) is to reconstruct the conductivity inside a physical body from boundary measurements of current and voltage at a set of contact electrodes. Almost all reconstruction algorithms for EIT assume the measurement setup is known precisely. In this work, the need for such prior geometric information is relaxed by considering algorithms that simultaneously reconstruct the conductivity, the electrode positions and the object shape. The introduced algorithms are tested via numerical experiments with both simulated and real-world data. As a specific application, imaging of stroke by EIT is discussed.

Lieu : Amphi L. Schwartz

• Homotopie en Géométrie Algébrique

### Mardi 22 janvier 14:00-15:00 - Alexis Roquefeuil - Univ. Angers

Invariants de Gromov—Witten $K$-théoriques et équations aux $q$-différences

Résumé : Notre point de départ sera les invariants de Gromov—Witten. Un exemple de tel invariant est le nombre de courbes rationnelles de degré $d>0$ dans le plan projectif complexe $\mathbbP^2_\mathbbC$ passant par $3d-1$ points. Ces invariants sont définis comme l’intersection de certaines classes cohomologiques qui vivent dans un espace de module en général assez singulier. En considérant les caractéristiques d’Euler de certains fibrés sur le même espace de module, A. Givental et Y.P. Lee ont défini en 2004 de nouveaux invariants qui constituent un analogue $K$-théorique des invariants de Gromov—Witten usuels. A partir de 2011, A. Givental et V. Tonita ont montré que ces invariants peuvent être exprimés par différents invariants de Gromov—Witten de nature cohomologique en utilisant un théorème ("virtuel") de Riemann—Roch.
Lorsque l’on se restreint aux courbes de genre 0, les invariants de Gromov—Witten cohomologiques peuvent être encodés dans un module différentiel appelé $\mathcalD$-module quantique. En essayant de faire la même chose du côté $K$-théorique, on trouve que des équations différentielles sont remplacées par des équations aux $q$-différences. Dans cet exposé, on s’intéressera aux liens entre la structure différentielle en cohomologie et la structure aux $q$-différences en $K$-théorie. Dans le cas des espaces projectifs, on cherchera à appliquer un procédé appelé confluence, qui consiste à faire "$q \to 1$" dans une équation aux $q$-différences pour obtenir une équation différentielle. Ce procédé permettra de passer de la K-théorie à la cohomologie pour la (petite) fonction $J$ de Givental.

• Homotopie en Géométrie Algébrique

### Mardi 22 janvier 15:30-16:30 - Joseph Tapia - IMT

Groupes formels I