Broadly speaking the group is focused around Partial Differential Equations and the Calculus of Variations.

Turbulence and the Euler equations

A fundamental problem of the theory of turbulence is to find a satisfactory mathematical framework linking the Navier-Stokes equations to the statistical theory of Kolmogorov. One of the cornerstones of the statistical theory is the famous 5/3 law, predicting the power law of the energy spectrum in turbulent flows. Although this law concerns the Navier-Stokes equations, a very closely related conjecture was made by Onsager regarding the critical Hölder regularity of weak solutions of the Euler equations which preserve the energy. Recently, in joint work with Camillo De Lellis, I have devoted a lot of effort in constructing weak solutions of the Euler equations at the critical 1/3 Hölder exponent. Based on "Mikado flows" and Phil Isett's gluing technique, we can now reach any uniform exponent less than 1/3.

Non-uniqueness and h-principle

A central difficulty in the Euler equations is the inherent non-uniqueness and pathological behaviour of weak solutions. This non-uniqueness, rather than being an isolated phenomenon, turns out to be directly linked to the celebrated construction of Nash and Kuiper of rough isometric embeddings and, more generally, to Gromov's h-principle in geometry. The same phenomenon appears in several other equations from fluid dynamics as well.

Quasiregular maps and Beltrami equations

Quasiconvexity and Morrey’s conjecture

Rank-one convexity

Rigidity

Polyconvexity and regularity in the Calculus of Variations