Johns Hopkins Math Colloquium

Abstract: The algebraic structure of the group of volume-preserving homeomorphisms of a manifold of dimension at least three has been well-understood since the work of Fathi from the 70s.  However, the surface case has long remained a mystery.  I will discuss joint work clearing up parts of this mystery.  Some Weyl type laws that have recently been established in low-dimensional symplectic geometry play a key role in the arguments.

Abstract: We study Brownian motion advected by a stationary, divergence-free random drift whose spatial correlations decay slowly. Such long-range dependence is expected to produce superdiffusion: for a typical realization of the drift, the variance of the displacement of the particle grows faster than linearly in time, with a precise exponent determined by the correlation structure of the drift. This behavior was predicted in the physics literature around 1990 via perturbative renormalization group heuristic arguments.

We recast the problem in PDE terms via the infinitesimal generator of the stochastic process, which is a divergence-form drift-diffusion operator, whose random coefficients exhibit an approximate self-similarity across scales. Our main tool is a scale-by-scale coarse-graining (renormalization) scheme. At each scale we compare the operator to an effective Laplacian with a diffusivity that depends on the scale, and we obtain quantitative control of the approximation error. Because of the self-similar structure, this procedure must be iterated across all scales: the operator does not converge to a single homogenized limit; instead, the analysis yields a scale-dependent effective diffusivity. This may be viewed as a rigorous counterpart of the earlier renormalization group predictions.

A key analytic input is an anomalous regularization phenomenon: we prove near-Lipschitz regularity for solutions that are uniform in the microscopic (molecular) diffusivity parameter, allowing quantitative control even in the regime of very weak underlying diffusion. In other words, the chaotic behavior of the random drift produces regularity in the solutions of the equation. 

The talk is based on joint work with Ahmed Bou-Rabee (UPenn) and Tuomo Kuusi (Helsinki) which is available here: https://arxiv.org/abs/2601.22142