Institut de recherche mathématique avancée

Résumé : We study a density-dependent Markov jump process describing a population where each individual is characterized by a type, and reproduces at rates depending both on its type and on the population type distribution. We are interested in the empirical distribution of ancestral lineages in the population process. First, we exhibit a time-inhomogeneous Markov process, which allows to capture the behavior of a sampled lineage in the population process. This is achieved through a many-to-one formula, which relates the expected value of a functional evaluated over the lineages in the population process to the expectation of the functional evaluated along this time-inhomogeneous process. This provides a direct interpretation of the underlying survivorship bias. Second, we consider the large population regime, when the population size grows to infinity. Under classical assumptions, the population type distribution converges to a deterministic limit. Here, we focus on the empirical distribution of ancestral lineages in this large population limit, for which we establish a many-to-one formula. Using coupling arguments, we further quantify the approximation error which arises when sampling in this large population approximation instead of the finite-size population process.

Résumé : Ocean circulation models cover a spatial domain up to the planetary scale. Due to computational constraints, numerical simulations therefore rely on relatively coarse meshes, with cell areas of the order of 10 000 km^2. As a consequence, finite volume schemes used for discretization induce a significant numerical entropy loss. This phenomenon leads to unphysical behaviours in the simulation result, such as the mixing of water masses with distinct temperatures and salinities. To address this problem, referred to as "spurious mixing" by oceanographers, localizing and quantifying this numerical entropy loss in high-order codes is crucial. A method was proposed by Aguillon, Audusse, Desveaux and Salomon, but is limited to the one-dimensional case.

I will introduce a method suitable for simulations in two dimensions of space, including high order schemes and in the presence of source terms. This approach relies on projecting a consistent flux onto the set of fluxes satisfying a discrete entropy inequality. The projection is carried out by an optimization algorithm with inequality constraints. Then, I will discuss the theoretical guarantees provided by the method, in particular the establishment of a Lax-Wendroff-type theorem, and conclude with numerical results for the shallow water equations.

This is a joint work with Nina Aguillon and Julien Salomon.

Résumé : Résumé : The Cremona group of rank N over a field K is the group of birational transformations of the N-dimensional projective space over K. A classical theorem by Noether and Castelnuovo states that the Cremona group of rank 2 over the complex numbers is generated by the birational involution sending (x,y) onto (1/x,1/y) and PGL(3). In particular, it is generated by involutions. Over non-algebraically closed fields, and even more so in higher dimensions, generators are more difficult to describe. The group theoretic properties of Cremona groups often depend on the rank and on the field. In this talk, I will tell the story about two theorems: A) The Cremona group of rank 2 over any perfect field is generated by involutions [joint with S. Lamy]. B) The Cremona group of rank at least 4 over the complex numbers admits the free group over an uncountable set as a quotient [joint with J. Blanc and E. Yasinsky]. Both of these theorems rely on the Sarkisov program, which gives an interesting set of generators not of the Cremona group but of a larger groupoid.

Résumé : Le but de cet exposé est de présenter certains résultats récents obtenus en vue du calcul des tables de caractères des modules de p-permutation de "petits" groupes finis et de la création d'une base de données de telles tables. On passera aussi en revue l'importance de telles classifications dans le contexte des équivalences de blocs d'algèbres de groupes finis.

Résumé : In this talk, I will give an introduction to a class of model reduction techniques for parametric partial differential equations, the so-called Reduced Basis (RB) method. Parameters can represent geometric configuration, physical properties, boundary conditions or source terms. These techniques are motivated by applications such as design, control or optimization: they provide low-dimensional and hence rapidly computable approximations for the solution. Important aspects are basis generation and certification of the simulation results by suitable a posteriori error control. I will close the presentation with a few of my research developments.

Résumé : The period-index conjecture is a fundamental problem concerning the Brauer group of algebraic varieties. For hyper-Kähler varieties, whose (birational) geometry is controlled by the second cohomology, it is expected that a stronger form of this conjecture holds. In this talk, I will present joint work with Daniel Huybrechts that provides new evidence for this expectation. Following a brief introduction to the problem, I will discuss a proof of a variant of the conjecture where the classical index is replaced by a Hodge-theoretic one. Then, I will explain how to verify the conjecture for most Brauer classes on hyper-Kähler varieties of K3n-type.