Abstract: Piecewise Deterministic Markov Processes (PDMPs) describe deterministic dynamical systems whose parameters undergo random jumps, making them versatile tools for modeling complex stochastic phenomena. Yet, simulating their trajectories can be computationally demanding. For a broad class of inference problems, an optimal sampling strategy can be characterized in terms of a generalized "committor function". We introduce a new adaptive importance sampling method designed to efficiently generate rare PDMP trajectories. The approach unfolds in two stages. First, in an offline phase, the PDMP is coarse-grained into a simpler graph-based process, enabling explicit computation of key quantities and yielding a low-cost approximation of the committor function. Then, in an online phase, trajectories are sampled from a distribution guided by this approximation and iteratively improved via cross-entropy minimization. We provide asymptotic guarantees for the method and demonstrate its effectiveness through the estimation of the failure probability of a complex industrial system.

Bayesian Inversion consists of deriving the posterior distribution of unknown parameters or functions from partial and indirect observations of a system. When the dimension of the search space is high or infinite, methods leveraging local information, such as derivatives of different orders, of the target probability measure have the advantages to converge faster than Monte-Carlo sampling techniques. Nevertheless, many applications are characterized by posterior distributions with low regularity or gradients that are intractable to compute. An interesting research direction consists in using interacting particle systems to explore the potential landscape, and Ensemble Kalman Sampler (EKS) is one of those. In this talk, we consider a simplified EKS dynamics, where the gradient of the potential is approximated by finite differences using independent Ornstein-Uhlenbeck processes that explore the neighborhood of the candidate parameter. We will characterize the invariant distribution of this system and compare its dynamics to the overdamped Langevin process.

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.

Symplectic fillings of lens spaces were classified by McDuff and Lisca in the early 2000s. A special class of these fillings arise as fillings of lens spaces L(p^2,pq-1), which admit symplectic fillings with vanishing second Betti numbers. In particular, they are symplectic models for rational homology balls B_{p,q}. We study symplectic embeddings of these models into CP2. We show that such embeddings exist if and only if p is a "Markov number" by "elementary" methods. This is joint work with N. Adaloglou, J. Brendel, J. Evans, and F. Schlenk.

Dans cette présentation, nous discutons de résultats asymptotiques, en particulier de variation régulière et de déviations larges, pour certaines classes de processus en clusters de Poisson et pour quelques fonctionnelles associées. L’accent est mis sur le cas du processus de Hawkes (marqué), qui sert de fil conducteur et d’exemple central. Nous illustrons ensuite la pertinence des hypothèses du cadre théorique par une application à des données sismologiques suisses. Il s’agit d’une travail conjoint avec Valérie Chavez-Demoulin et Olivier Wintenberger.

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