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.