I will start with a motivation on how the Hasse principle naturally arises. I will then explain the notion of the Brauer--Manin obstruction to the Hasse principle and may mention along the way how it can prevent adelic approximation (weak or strong depending on the topology). I will then explain how to adapt Manin's classic construction to integral points. If time permits I will also explain the semi-integral (Campana and Darmon) version of that obstruction. I will conclude with a calculation of an explicit example where the Hasse principle fails because of a Brauer-Manin obstruction.

Specialization theorems of Néron and Silverman imply that in a non-isotrivial family of abelian varieties over a number field, all but a sparse set of members have Mordell-Weil rank at least the generic rank of the family. A natural question follows: how large is the set of members whose rank is strictly larger? In this talk we will explore this question for the relative jacobian of a pencil of hyperelliptic curves on a K3 surface.

We will introduce the Chow ring of moduli space M_g,n and discuss my work which proves the rational Chow rings of M_5,8 and M_5,9 are tautological.

Sampling from heavy-tailed and multimodal distributions is challenging when neither the target density nor the proposal density can be evaluated, as encountered in α-stable Lévy-driven fractional Langevin algorithms. While the target distribution can be estimated from data using score-based or energy-based models, the α-stable proposal density and its score are generally unavailable, rendering classical density-based Metropolis-Hastings (MH) corrections impractical. Consequently, existing fractional Langevin methods operate in an unadjusted regime, often exhibiting substantial finite-time errors and poor empirical control over tail behavior. To address these limitations, we first introduce a general, fully score-based MH adjustment mechanism. Building upon this framework, we develop the Metropolis-Adjusted Fractional Langevin Algorithm (MAFLA). MAFLA employs designed proxies for fractional proposal score gradients under isotropic symmetric \alpha-stable noise and learns the acceptance function via Score Balance Matching. We empirically demonstrate the strong performance of MAFLA across a series of tasks, including combinatorial optimization problems, where the method significantly improves finite-time sampling accuracy over unadjusted fractional Langevin dynamics.

Generating extreme events poses two distinct obstacles, and generative models may easily fail at both. The difficulty is not confined to heavy tails, though those are the hardest case, and in the multivariate setting different directions may sit in different tail regimes, none of them known in advance. The first obstacle is in the generative objective itself, where every standard choice imposes regularity on the measures: KL and f-divergences require absolute continuity and can easily fail on singular or heavy-tailed targets, while Wasserstein distances require moments of both arguments and may not have well-defined variational derivatives. Lipschitz-regularized divergences, by contrast, are infimal convolutions of a divergence with a transport cost, i.e. a proximal regularization in the Wasserstein-1 geometry, and inherit the best of both, and more: the only assumption needed is on the source, which we design (a finite first moment), and, most importantly, none at all on the target, singular or heavy-tailed alike. The variational derivatives then exist, are finite and unique, and the Wasserstein gradient flow is well posed on targets other objectives cannot handle. Its dissipation inequality also yields a principled, data-driven stopping criterion in place of a preset number of steps. The second obstacle is that the transport itself cannot be regular for learning heavier tails than the source's. A Lipschitz map sends a light-tailed source to a light-tailed output, so a well-behaved flow never reaches the tail. CVaR, which measures tail risk, is blind to the bulk and aggregates the whole tail into a single scalar observable, so relatively few tail samples suffice to estimate it reliably, where densities and their gradients cannot be resolved at all; adding it to the generative objective - the two terms together forming a free energy - revives the transport in the sparsely sampled tail, through a bounded but non-Lipschitz velocity. The resulting correction is tail-agnostic and applies to the samples of any pre-trained model without access to its architecture; across synthetic and real target data it sharply improves both bulk and tail accuracy over pre-trained baselines and over recent tail-adapted GANs, diffusions and normalizing flows.

Extreme events in complex dynamical systems may arise from distinct mechanisms, but identifying their drivers and determining whether models represent the relevant pathways remain major challenges. This talk presents a unified information-propagation perspective on these two questions. First, assimilative causal inference combines causal diagnostics with data assimilation to identify the variables and dynamical pathways that contribute to an extreme event and to quantify their ranges of influence. Examples involving both noise-driven and nonlinearity-driven extremes illustrate how different mechanisms produce distinct causal signatures and precursor structures. Second, controlled state reconstruction is used to interrogate whether a dynamical model transmits observational information toward an extreme-event target through physically meaningful pathways. The resulting reconstruction fingerprints reveal not only whether additional observations improve the representation of an extreme, but also where information transfer is distorted or suppressed. Applications to ENSO extremes in CMIP models examine the coupled pathways linking sea-surface temperature, winds, thermocline variability, and ocean recharge. These methods aim to move beyond predicting extreme events toward understanding their mechanisms and assessing whether a model’s explanation of them can be trusted.

Stochastic optimal control (SOC) has application in many areas, including rare events, economics, and engineering. In this talk, we will present a highly parallel and derivative-free martingale neural network method, based on the probability theory of Varadhan’s martingale formulation of PDEs, to solve Hamilton-Jacobi-Bellman (HJB) equations arising from SOC problems in very high dimensions. The HJB equation is reformulated into a martingale problem such that loss functions will not require the computation of the gradient or Hessian matrix of the PDE solution and can be computed in parallel in both time and spatial domains. Moreover, the martingale conditions for the PDE are enforced using a Galerkin method realized with adversarial learning techniques, eliminating the need for direct computation of the conditional expectations associated with the martingale property. Also, a derivative-free implementation of the maximum principle for optimal controls is introduced. Numerical results demonstrate the effectiveness and efficiency of the proposed method, which is capable of solving HJB accurately and fast in dimensions as high as 10,000.

The quasipotential is an analog of the potential function for nongradient SDEs. It determines the exponential factors in transition rates and invariant measures and defines the most probable escape paths from the basins of attractors. Its computation in large regions surrounding the system’s attractors requires solving an optimal control problem on a grid. I will discuss two approaches for computing the quasipotential. The first one comprises Dijkstra-like ordered line-integral methods (OLIMs), including the latest version suitable for dimensions up to three and variable, anisotropic, and rank-deficient diffusion. The second one traces the unstable Lagrangian manifold of the associated Hamiltonian system. I will demonstrate the application of these approaches to problems coming from ecology, climate science, and engineering.

Long-time rare-event statistics can be estimated from ensembles of trajectories that are much shorter than the event timescale. Committors, mean first passage times, and related rare event statistics all solve Bellman equations defined in terms of stopped trajectories. Markov state models are Galerkin approximations of these equations. On finite-state spaces we can establish relative error bounds for these quantities that are nearly independent of a condition number or related quantity (return-time, spectral gap, etc), partially explaining the empirical success of these methods on rare event problems (where condition numbers are often extremely large). I'll also describe recent approaches to non-linear parametrization of these equations as well as their failure, in the case of state-of-the-art AI weather emulators, when training data does not sufficiently cover dynamically important regions of state space and show that combining parametrized approximation of rare event statistics with rare-event simulation is one promising route to generating the necessary training data.