Critical loop O(n)O(n) models are conjectured to be conformally invariant in the scaling limit—a striking bridge between random geometry and conformal field theory. In this talk, we investigate connection probabilities for loop O(n)O(n) models in polygons, which can be predicted through solutions of the Belavin–Polyakov–Zamolodchikov (BPZ) equations. This conjecture has been verified for several fundamental models: the critical Ising model, the FK-Ising model, percolation, and the uniform spanning tree. We then present a complete classification of the linear solution space of the BPZ system, showing how it decomposes under the action of three conformal Ward identities. Beyond the linear regime, we analyze the semiclassical limit of the BPZ system, which takes the form of a nonlinear Hamilton–Jacobi system, and determine the number of global real-valued solutions. This project lies at the interface of random interfaces and deterministic geometry, drawing on tools from stochastic calculus, PDE analysis, and Schubert calculus.

I will discuss some properties of the space of subgaussian vectors, and how they relate to the geometry of large sets in Gaussian spaces and the convexity conjecture of M. Talagrand. This is partly joint with Merrick Hua and Stefan Tudose.

Chinburg and Reid initiated using cross ratios to detect self-intersections of closed geodesics in hyperbolic 3-manifolds and construct closed hyperbolic 3-manifolds all of whose closed geodesics are simple. Since then, a natural question is whether every closed hyperbolic 3-manifold contain infinitely many simple closed geodesics. Wu and Xue ask whether random closed geodesics on hyperbolic surfaces are filling. Inspired by their probabilistic perspective, we develop a simple polynomial method, which shows that in a finite-volume hyperbolic n-manifold of dimension at least 3, random closed geodesics are simple. This is a joint work with Bohan Yang.