NCAFA

Objective:

Since its inception by Joseph Fourier in the 1820s to solve the problem of the propagation of heat, Fourier analysis has been a fundamental mathematical tool to understand plenty of physical phenomena, from the most classical problems related to mechanical waves to quantum mechanics. Our objective with this project is to iterate on the search of new directions and applications of Fourier analysis by exploiting and strengthening its connections to fundamental physical phenomena whose mathematical study has recently seen a surge of activity, like the turbulence of fluids, the turbulence of waves, optical effects like the Talbot effect and questions arising from statistical mechanics. We propose to divide our proposal in four Working Packages, each one looking for applications of Fourier analysis in a different direction: WP1. Questions from fluid turbulence. We will investigate the evolution of vortex filaments and their relationship with the Navier-Stokes equations, and the concepts of multifractality and intermittency. WP2. Questions from wave turbulence. We will work on the rigorous derivation of wave kinetic equations for dispersive PDE, on the invariance and quasi -invariance of Gibbs and gaussian measures, and on their applications. WP3. Uncertainty principles and uniqueness properties. We will study dynamical uncertainty principles for Schrödinger and Helmholtz equations related to Talbot effects, and uniqueness properties of non-local dispersive equations. WP4. Other related problems in Fourier analysis. We will study interesting and relevant versions of the Carleson convergence problem, among which we highlight the higher-order, the fractal, the sequential and the directional problems.