Specific programme: European Research Council Advanced Grant
UPV/EHU Partner Status: Coordinator
UPV/EHU PI: Luis Vega
Project start: 01/12/2015
Project end: 30/11/2020

Brief description: This project sets forth cutting-edge challenges in the field of Mathematical Physics that will be solved within a common framework by making novel use of classical tools of Harmonic Analysis such as Oscillatory Integrals and Trigonometric Sums, the Cauchy operator, and the so-called Carleman estimates. Three aspects will be covered: 1.Vortex Filament Equation (VFE) 2.Relativistic and Non-relativistic Critical Electromagnetic Hamiltonians 3.Uncertainty Principles (UPs) and Applications The interaction of vortex filaments is considered a key issue in order to understand turbulence which is seen by many as the most relevant unsolved problem of classical physics. VFE first appeared as an approximation of the dynamics of isolated vortex filaments. The research will investigate what happens when at time zero the filament is a regular polygon. Preliminary theoretical arguments together with some numerical experiments suggest that the different corners behave like different vortex filaments that interact with each other in such a way that the dynamics seem chaotic. The so-called Frisch-Parisi conjecture will be proven, showing that behind this chaotic behavior there is an underlying algebraic structure that controls the dynamics. The Dirac equation, despite being one of the basic equations of Mathematical Physics, is very poorly understood from an analytical point of view. The classical Cauchy operator will be used in a modern way to explain some key Hamiltonian systems such as the MIT bag model for quark confinement. UPs are at the heart of different fields like Quantum Mechanics, Harmonic Analysis, and Information Theory. A new approach will be used to analyze modern versions of UPs that are not well understood. In order to do this, the problem will be looked at from the point of view of partial differential equations making novel use of the Carleman estimates. This analysis will also be extended to the discrete setting where even classical UPs such the one by Hardy are not solved yet.