The celebrated Euler and Navier-Stokes equations of fluid dynamics model the motion of particles in an inviscid and in a viscous fluid, respectively. Their mathematical understanding poses formidable challenges, mainly stemming from the presence of irregular solutions. The behavior of these irregular solutions has, in turn, fascinating and still not fully understood connections with physical problems involving turbulent phenomena. In this area, some recent breakthrough results constitute a step forward: for instance, the proof of the Onsager conjecture in the framework of the Kolmogorov theory of turbulence and certain quantitative bounds on mixing rates.

The goal of Maria Colombo and of her research team is to provide new insight on the theoretical understanding of irregular solutions, answer fundamental questions about (non)-uniqueness of solutions and typicality of nonsmooth behaviors.

Continuous optimization is a cornerstone of technology: we use it to transform vast amounts of data into actionable information, be it by solving inverse problems in imaging, performing statistical estimation in signal processing or training neural networks in machine learning. To realize the full potential of these computational tools, we need to push our understanding of the limits and opportunities of optimization beyond the classical mathematical framework (convexity).

With the project GEOSYM, the goal of Nicolas Boumal and his team is to harness other mathematical structures that are ubiquitous in applications, namely, geometry and symmetry, to tackle non-convex optimization reliably.