2026, March 20

This session was held at Imperial College London, Huxley building, room 340.

Schedule:

10:00-11:00

Nikolaos Zygouras (University of Warwick): From subcritical to critical Singular SPDEs

The past decade has witnessed a revolution in the treatment of singular stochastic partial differential equations (SPDEs) through mainly Hairer’s Theory of Regularity Structures, Gubinelli-Imkeller-Perkowski Paracontrolled Distributions and also via renormalisation approaches. These theories, however, are limited by the so-called “criticality” or “critical dimension”. The first steps towards criticality are now shyly being made. After giving a general taste of singular SPDEs, I will present some of the first steps at criticality and discuss in particular the Critical 2d Stochastic Heat Flow, which was constructed jointly with Francesco Caravenna and Rongfeng Sun, as a non-trivial solution of the Stochastic Heat Equation in dimension 2.

11:30-12:30

Rita Teixeira da Costa (University of Cambridge): On the stability of rotating black holes

I will discuss some recent work with collaborators on the stability of rotating black hole solutions to the Einstein vacuum equations.

14:30-15:30

Scott Armstrong (CNRS and Sorbonne Université): Anomalous diffusivity and regularity for random incompressible flows

We consider the behavior of Brownian motion in a "turbulent" drift, that is, a stationary, incompressible random drift field with slowly decaying correlations. In this setting, one expects the variance of the displacement to grow faster than linearly in time, with an exponent determined by the correlation structure of the drift. This behavior was predicted by physicists in 1990 using perturbative renormalization group heuristics. We formulate the problem as a PDE problem via the associated divergence-form drift–diffusion operator, which has self-similar (or multifractal) coefficients. We apply a scale-by-scale coarse-graining scheme to this operator. At each scale, this produces an effective Laplacian whose diffusivity depends on the scale, together with quantitative control of the approximation error. In other words, we use methods originating in quantitative homogenization theory, but we must iteratively perform infinitely many homogenizations, and the operator never “finishes” homogenizing because of its self-similar structure. This may be viewed as a rigorous version of the perturbative RG heuristics. A crucial role is played by anomalous regularization, that is, regularity estimates for solutions that are independent of the bare molecular diffusivity. The work I will describe is based on our joint paper with A. Bou-Rabee and T. Kuusi.

16:00-17:00

Giada Franz (Université Gustave Eiffel): Morse theory for minimal surfaces: emanating flow lines

Morse theory provides a framework for studying the infinite-dimensional space of embedded hypersurfaces in a fixed ambient manifold through the analysis of the area functional. A special role in this picture is played by minimal hypersurfaces, which are precisely the critical points of this functional. By analogy with classical Morse theory, one expects that a minimal hypersurface of Morse index I should admit an I-dimensional unstable manifold emanating from it. In this talk, I will present recent results that make this heuristic precise by constructing and characterizing ancient solutions to mean curvature flow that emanate from minimal hypersurfaces. Particular focus will be given to the case of hypersurfaces with boundary.

Previous sessions

List of titles and abstracts here.