We consider the Navier-Stokes equations with constant/variable density in a 2D slab domain, horizontally periodic.  We will study those two nonlinear perturbed problems around special equilibrium, under which Navier-slip boundary conditions are imposed, with constant slip coefficients. In this talk, the influence of slip coefficients on linear instability will be presented to obtain infinitely many normal mode solutions, following the operator method of Lafitte and Nguyen. Hence, we prove the nonlinear instability with a wide class of initial data, thanks to the abstract frameworks of Guo-Strauss and of Desjardins-Grenier. 

We introduce a novel statistical framework for the analysis of replicated point processes that allows for the study of point pattern variability at a population level. By treating point process realizations as random measures, we adopt a functional analysis perspective and propose a form of functional Principal Component Analysis (fPCA) for point processes. The originality of our method is to base our analysis on the cumulative mass functions of the random measures which gives us a direct and interpretable analysis. Key theoretical contributions include establishing a Karhunen-Loève expansion for the random measures and a Mercer Theorem for covariance measures. We establish convergence in a strong sense, and introduce the concept of principal measures, which can be seen as latent processes governing the dynamics of the observed point patterns. We propose an easy-to-implement estimation strategy of eigenelements for which parametric rates are achieved. We fully characterize the solutions of our  approach to Poisson and Hawkes processes  and validate our methodology via simulations and diverse applications in seismology, single-cell biology and neurosiences, demonstrating its versatility and effectiveness.

We study the propagation of coherent states in self-interacting bosonic quantum field theories in the semi-classical (mean-field) regime. Our work focuses in particular on the spatially cutoff P(\phi)_2 model, under standard assumptions ensuring essential self-adjointness of the Hamiltonian. For this model, the classical field equation is given by a non-linear Klein-Gordon equation, whose well-posedness constitutes the first step of our analysis. We then apply Hepp’s method, combined with a detailed study of both the classical and quantum dynamics, in order to construct an asymptotic expansion of arbitrary order for the quantum evolution of coherent states.

Découvert en 1959, le théorème de Dye stipule qu'à identification mesurable près, toutes les bijections ergodiques préservant une mesure de probabilité (sans atomes sur un espace standard) induisent la même partition de l'espace en orbites. En 1969, de l'autre côté du rideau de fer, Belinskaya démontre qu'à l'inverse, si l'on demande que l'identification des orbites soit intégrable (en un sens que l'on précisera), alors les deux bijections sont conjuguées, quitte à éventuellement remplacer l'une par son inverse. Dans cet exposé, je présenterai le théorème de Belinskaya ainsi que quelques groupes polonais associés à de la dynamique mesurable, groupes dont l'étude est motivée par ce théorème. Il s'agit d'une part des groupes pleins L1, que j'ai définis et étudiés en 2018, et d'autre part les groupes pleins commensurants, qu'Antoine Derimay a étudiés lors de son stage de M2 sous ma direction en 2024.