While there is a deep mathematical understanding of the 2D Ising model, its continuous $S^2$-valued counterpart—the classical Heisenberg model—remains one of the most significant challenges in mathematical physics. In 1975, Polyakov predicted that this system remains highly decorrelated at all positive temperatures due to its non-abelian symmetry. In this talk, I will discuss the behavior of this model under small perturbations of the sphere's geometry. I will highlight how this problem naturally connects to questions in analysis, such as the properties of harmonic maps. 

This is a joint work with Nathan de Montgolfier.