Analysis and Probability (AP)

Randomness in materials.

Often, the effective conductivity or the effective elastic moduli of a heterogeneous material just depend on the statistics of its microstructure. The effective behavior can be numerically inferred from representative images or computer-generated samples if the ensemble is known. The fluctuation and bias of these methods are being reliably estimated, an instance of uncertainty quantification. This subject of upscaling or homogenization is inspired by studying more prototypical models of random geometries, like percolation clusters, and their large-scale behavior, for instance, probed by random walks. The supposedly universal behavior at criticality is a major challenge.

While for quenched noise, the challenge resides in capturing self-averaging on large scales, the challenge of thermal noise in continuum models lies in taming its unprotected interaction with nonlinearities on small scales, which requires a renormalization. A prototypical example is provided by quantum field theory, and a mathematically rigorous treatment is emerging, thanks to the theory development of Fields medalist Hairer. But also models in mesoscopic physics, like fluctuating hydrodynamics, come with similar and unresolved challenges.

Roughness in fluids.

Recently, our confidence in the predictive power of conservation laws in continuum mechanics has been shaken. It has been shaken by the construction of an abundance of rough solutions to the partial differential equations of inviscid and viscous fluid dynamics, the Euler and Navier-Stokes equations. Despite their non-uniqueness, some of these solutions carry a physical meaning in turbulence: they allow for energy dissipation while conserving mass and momentum, as predicted by L.~Onsager in 1949. Also, they reflect many of the scaling properties predicted to hold in the inertial range by Kolmogorov's 1941 statistical theory.

The method of construction, closely related to Nash's and Gromov's work on the h-principle, has led to a surge of activity both on the purely mathematical and the applied fluid dynamical side. Concerning the former, the construction by T. Buckmaster and V. Vicol of mild solutions of the Navier-Stokes equations outside the Leray-Hopf class has led the community to reevaluate our understanding of the nature of supercritical PDE completely. Concerning the latter, the technique led to a number of advances in understanding the "infinite Reynolds number turbulence limit" in various contexts, such as the role of helicity in turbulent plasmas in connection with the existence of a Taylor-Woltjer final state or the evolution of unstable interfaces in fluids such as the motion of vortex sheet.

Beyond differential geometry.

The community is debating robust substitutes for the analytic notions of curvature from differential geometry adapted to rough settings. There are several settings where roughness appears: when the underlying space is discrete (e.g., large graphs) or noisy (e.g., random surfaces); when one is interested in the critical regularity connected to geometric rigidity, as in the case of CR structures or isometric immersions.

Some of these synthetic and metric notions of curvature are motivated by optimal transportation with its connection to shape recognition and manifold learning. Optimal transportation, to which the recent Fields medals for Villani and Figalli are connected, is a topic that intertwines convex optimization, statistics, and fully nonlinear partial differential equations.

It gives rise to a metric and curvature on the space of probability measures; diffusion can be seen as the steepest descent of the entropy in this infinite-dimensional landscape, an image that has been taken up in the analysis of stochastic gradient descent. The regularity theory of this variational problem can now be assimilated to one for minimal surfaces and allows to capture the mesoscopics of matching random point clouds, in line with the heuristics of Parisi.