Riemann surface is a closed surface equipped with a complex structure. If the genus is at least two, then equivalently it is a closed hyperbolic surface. We'll give an overview of some of the properties of the moduli space of such surfaces and their mapping class group, and we connect geometric information to the construction of geometric representations of the mapping class group on suitably chosen $L^p$-spaces and discuss some open questions.

A central problem in statistical physics is to understand how spins placed on the lattice $Z^d$ interact and collectively organize at different temperatures. When the spins take values in a discrete set — for instance in the celebrated Ising model, where $\sigma_x\in\{−1,+1\}$ — the mechanisms governing phase transitions are by now relatively well understood. The situation changes dramatically when the spins take values in a continuous space, such as the unit circle $S^1$ in the XY model or the unit sphere $S^2$ in the classical Heisenberg model. In this setting, new phenomena appear, and the behavior depends strongly on whether the underlying symmetry is Abelian or non-Abelian. In particular, the non-Abelian case remains far more mysterious. In this talk, I will introduce the mathematics of spin systems with continuous symmetry, emphasizing their deep connections with analysis, including harmonic functions, harmonic maps, and geometric analysis. I will also describe some recent results and open problems in the area. No prior background in statistical physics or probability will be assumed. Based on joint works with J. Aru, D. van Engelenburg, P. Dario, N. de Montgolfier, A. Sepúlveda and T. Spencer.

Many data sets arising in stochastic systems, dynamics, and time series are naturally represented by paths rather than by points in a finite-dimensional space. The signature transform, which associates to a path a hierarchy of iterated integrals, provides a powerful way of encoding such objects while retaining rich algebraic and analytic structure. In this talk, I will introduce the basic ideas behind signatures and explain how they can be used for learning problems on path space. I will focus in particular on stochastic regression and on signature kernels, which allow one to work with the signature representation without computing its infinitely many coordinates explicitly. Along the way, I will highlight connections between rough path theory, kernel methods, partial differential equations, and data-driven learning.

Wide neural networks may be viewed as high-dimensional interacting particle systems: each neuron evolves during training, while interacting with the others through empirical averages determined by the network output and the loss. This perspective naturally suggests a mean-field limit as the width tends to infinity. However, in contrast to classical mean-field models, the relevant scaling for feature learning is delicate. The $\mu P$ parametrisation is designed to produce nontrivial infinite-width training dynamics, avoiding the purely linearised behaviour of the neural tangent kernel regime. In this talk, I will discuss a new decoupling result for neural network training in the $\mu P$ setting. Starting from Gaussian initialisation, we show that the original interacting neuron system can be explicitly coupled to a completely decoupled system. In this decoupled system, individual neurons evolve independently of each other according to effective one-particle dynamics. They interact with the rest of the network only through deterministic mean-field quantities. The result can be understood as a propagation-of-chaos statement adapted to the $\mu P$ scaling, with an explicit coupling between the true dynamics and the decoupled system.

It is well-established that Patterson-Sullivan measures on the boundary of a hyperbolic space, along with the associated Bowen-Margulis-Sullivan measure, provide valuable insights into the action of a group of isometries on the space's boundary through analysis of the geodesic flow. Given that paths of a random walk on a hyperbolic groups lie close to the group's quasi-geodesics, it is natural to ask whether similar behaviour can also be seen in the flow along bi-infinite random walk paths. In this talk, I will show how studying bi-infinite random walks on a discrete hyperbolic group $G$ leads to an analogue of the Patterson-Sullivan measure on $\partial^2G$. This measure can be constructed in multiple measure-equivalent ways, each giving distinct perspectives on its intrinsic structure. Moreover, as in the classical case, the action $G \curvearrowright \partial^2 G$ is ergodic with respect to this measure. Central to the construction of these measures is the almost sure convergence of the random walk to the boundary $\partial G$ and the study of the distribution of the hitting points. This talk is based on joint work with Mahan Mj and Chiranjib Mukherjee.

The uniqueness of the Kerr-de Sitter family of black hole spacetimes as stationary solutions to the Einstein vacuum equations is a crucial ingredient to understanding the final states of positive cosmological constant universes, such as our physical universe. In the asymptotically flat case, Kerr was shown to be the unique analytic stationary solution to the Einstein vacuum equations via a combination of results by Hawking, Carter, and Robinson. Outside of analyticity, Alexakis, Ionescu, and Klainerman showed several conditional rigidity results for Kerr. In this talk, I will discuss some recent work in the spirit of Alexakis, Ionescu, and Klainerman showing the uniqueness of the stationary region of Kerr-de Sitter within the smooth class of stationary solutions to the Einstein vacuum equations with a positive cosmological constant.