Liouville quantum geometry and turbulent flows – Liouville

This research project is organized along two seemingly unrelated directions: 2d quantum gravity and 3d turbulence.

(1) Mathematical Liouville gravity deals with the geometry of large random planar maps. Historically, conformal invariance was a key ingredient in the construction of Liouville gravity in the physics literature. Conformal invariance has been restored recently with an attempt of understanding large random combinatorial planar maps once conformally embedded in the plane. A recent striking conjecture states that the geometry induced by these embeddings is asymptotically described by the exponential of a highly oscillating distribution, the planar Gaussian Free Field. Another important open problem in this field is to extract a "Liouville metric" out of the exponential of this Gaussian Free Field. The first major goal of our project is to make significant progress on these conjectures. We will combine for this several tools such as Gaussian multiplicative chaos, Liouville Brownian motion, planar maps, circle packings, QLE processes and Bouchaud trap models. This field is currently extremely active both in probability theory and in mathematical physics. Completing the above program would indeed provide a setup for a rigorous understanding of the celebrated KPZ relation by building a bridge from Liouville quantum gravity to 2d euclidean statistical physics.


(2) 3d turbulence. A more tractable ambition than solving Navier-Stokes equation is to construct explicit stochastic vector fields which combine key features of experimentally observed velocity fields. We make the mathematical framework precise by identifying four axioms that need to be satisfied. It has been observed recently that the Gaussian multiplicative chaos theory, once generalized to the matrix-valued case, could be used to create such a realistic velocity field. The extension to the matrix valued case is motivated by one of the main mechanism entering in the dynamics of the Navier-Stokes equations, namely vorticity stretching. As in 2d quantum gravity, the difficulty also lies in defining rigorously the exponential of 3d log-correlated distributions (or rather matrix-valued distributions). Even though new mathematical difficulties arise in this setting, we wish to complete this program by relying on the techniques that have been developed in the above context of 2d quantum gravity. If we succeed, one would thus construct the first genuine stochastic model of turbulent flow in the spirit of what Kolmogorov was aiming at.

This project combines several top level experts in a variety of fields which all together should enable us to come up with new results and new insights on these exciting conjectures.