A central question in the topology of distributions is whether there exist maximally non-integrable distributions beyond contact structures for which the h-principle fails, i.e. whether there are genuinely new invariants not dictated by algebraic topology. Most known classes instead satisfy strong flexibility results and are classified by their obvious homotopical data. In this talk I will briefly survey this landscape and then present a recent joint result with Álvaro del Pino and Eduardo Fernández giving a first example of rigidity beyond the contact world: a class of fat (corank- 2) distributions that naturally generalise contact structures, together with their adapted submanifolds, the prelegendrians. Using a canonical contactisation, we construct families of prelegendrians detected by contact-topological invariants, thereby showing that the ℎ h-principle fails in this setting.

TBA

Rediscovered by J-L LODAY in 1990's, Leibniz algebras are non-anticommutative versions of Lie algebras. In the last three decades numerous papers and results on Leibniz algebras appeared. Leibniz algebras play an important role in different areas of mathematics and physics. In this talk, Lie algebras and Leibniz algebras will be considered from a purely algebraical point of view. In this talk, we will give an introduction to Leibniz algebras, their bimodules and their cohomology. If we have time left, we will describe a special class of Leibniz algebras : the cyclic Leibniz algebras.

The full regularity of harmonic maps from a given surface into an arbitrary Riemannian manifold has been proved by Hélein in 1991. This is not true anymore when the domain has dimension strictly greater than 2, Rivière constructed an example of harmonic map from a 3-dimensional domain which is everywhere discontinuous in 1995. There are many possible generalizations of these maps to the higher dimensional case in order to recover the regularity of some "optimal" maps. For most of these generalizations, the optimal regularity in full generality is still open. In this talk, we will discuss some recent progress obtained for n-harmonic maps. This is a joint work with Armin Schikorra.

Quantum mechanics is well approximated by classical physics when Planck's constant is considered small, i.e., in the semi-classical limit. Typically, one can study an observable associated with a particle, such as its momentum or its position, and show that its dynamics is given by classical dynamics at first order, with corrections of the order of Planck's constant. In this talk, I will present more precisely the concept of semi-classical limits, the standard mathematical results known for non-relativistic quantum mechanics, and my work that concerns the semi-classical limit in the context of relativistic quantum mechanics. Concretely, I will show how to adapt the modulated energy method to the Klein-Gordon and Klein-Gordon-Maxwell equations and how to recover relativistic mechanics (instead of classical mechanics) at the semi-classical limit

We consider a tagged particle in mean field interaction with a Rayleigh gas of density N, and prove the convergence of its trajectory, as N goes to infinity, to the one of a diffusion process associated with the linear Landau equation. This is joint work with T. Bodineau.

The problem of testing linear hypotheses for the means of random functions is considered. This includes checking if the mean is zero, checking if two sample means are the same, and checking if the two means have a constant difference or ratio. The random function is defined on a multidimensional compact domain and several independent realizations are observed at random design points, possibly with heteroscedastic error. The number of design points of each realization of the random function can be bounded or arbitrarily large. For two-sample tests, the samples are allowed to be unbalanced and dependent. The testing approach is based on a non-asymptotic Gaussian approximation bound for the estimated Fourier coefficients. A pivotal chi-square type statistics is proposed.