We study model-free policy learning for discrete-time mean-field control (MFC) problems with finite state space and compact action space. In contrast to the extensive literature on value-based methods for MFC, policy-based approaches remain largely unexplored due to the intrinsic dependence of transition kernels and rewards on the evolving population state distribution, which prevents the direct use of likelihood-ratio estimators of policy gradients from classical single-agent reinforcement learning. We introduce a perturbation scheme on the state-distribution flow and prove that the gradient of the resulting perturbed value function converges to the true policy gradient as the perturbation magnitude vanishes. This construction yields a model-free estimator based solely on simulated trajectories and an auxiliary estimate of the sensitivity of the state distribution. Building on this framework, we develop MF-REINFORCE, a model-free policy gradient algorithm for MFC, and establish quantitative bounds on its bias and mean-squared error.

Calibrated fibrations are expected to play an important role in exceptional holonomy, much as special Lagrangian fibrations do in the SYZ picture for Calabi--Yau manifolds. Singular fibres are expected to be essential, and a natural question is whether they are forced by the geometry. In his PhD thesis, Baraglia showed that coassociative fibrations of compact full-holonomy G_2-manifolds must have singular fibres. The analogous problem for Cayley fibrations remained open for many years and turns out to be substantially more difficult, with its resolution relying on deep results from 4-manifold topology.

The proof takes some unexpected twists and turns, involving a Diophantine equation arising from the Spin(7)-structure, families Seiberg--Witten theory and parametrized homotopy theory. After a short crash course on Spin(7) geometry, I will explain how these pieces fit together. This is joint work with Jianfeng Lin and Viktor Majewski.

to follow

Control policies optimized in simulation can perform poorly in the real system when the simulator's parameters are estimated from limited data but the resulting parameter uncertainty is not represented inside the simulation. A common way to incorporate such ambiguity is to simulate each trajectory under a randomly drawn and unobservable parameter value. This forces the policy to act robustly at the start of the control episode. Over time, however, the policy can often infer the parameter value from its observations and specialize its actions accordingly, which is undesirable in systems where latent factors are expected to shift. In financial markets, for example, a policy hedging a derivative payoff should remain robust to changes in the volatility regime. To induce such continual robustness, we propose training policies in simulators where ambiguity can vary with the system's state but does not systematically vanish over time. We formalize this as stationary ambiguity: the simulator induces a stationary filter process over the latent state. We show how to construct such simulators and demonstrate, on hedging problems, that policies trained under stationary ambiguity maintain robustness to latent factors over time, leading to strong performance on real market data. As a modeling principle, stationary ambiguity informs which models make realistic simulators, how their parameters should be randomized, and how simulator and policy should be initialized. While our experiments focus on hedging, stationary ambiguity may also be useful for other control problems driven by exogenous stochastic processes with shifting latent structure.

This is a joint work with Amira Akkari, Ben Wood, and Lukas Gonon.

Recently, data-driven methods based on path signatures have gained prominence in mathematical finance. They rely on universal approximation theorems stating that continuous functionals on path space can be approximated uniformly on compact sets by linear functionals of the signature. In financial applications, this has led to the use of Stratonovich-signatures, although Itô integration is often the natural modeling framework.

In this talk, we establish a universality result for signatures of non-geometric rough paths. By augmenting the path with its rough path bracket, we obtain a quasi-shuffle structure that provides the algebraic basis for universality. For continuous semimartingales, this yields a universal approximation property for Itô signatures.

This talk is based on joint work with A. P. Kwossek and D. J. Prömel.