Objective 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 ApplicationsThe 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. I want to understand 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. I will prove the so-called Frisch-Parisi conjecture, 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. I will use the classical Cauchy operator 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. We want to use a new approach to analyze modern versions of UPs that are not well understood. In order to do this, I will look at the problem 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 Fields of science engineering and technologynanotechnologynano-materialstwo-dimensional nanostructuresgraphenenatural sciencesphysical sciencesquantum physicsnatural sciencesmathematicsapplied mathematicsmathematical physicsnatural sciencesphysical sciencestheoretical physicsparticle physicsquarksnatural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations Programme(s) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Topic(s) ERC-ADG-2014 - ERC Advanced Grant Call for proposal ERC-2014-ADG See other projects for this call Funding Scheme ERC-ADG - Advanced Grant Coordinator UNIVERSIDAD DEL PAIS VASCO/ EUSKAL HERRIKO UNIBERTSITATEA Net EU contribution € 903 352,60 Address Barrio sarriena s n 48940 Leioa Spain See on map Region Noreste País Vasco Bizkaia Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 1 672 103,00 Beneficiaries (2) Sort alphabetically Sort by Net EU contribution Expand all Collapse all UNIVERSIDAD DEL PAIS VASCO/ EUSKAL HERRIKO UNIBERTSITATEA Spain Net EU contribution € 903 352,60 Address Barrio sarriena s n 48940 Leioa See on map Region Noreste País Vasco Bizkaia Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 1 672 103,00 Third-party Legal entity other than a subcontractor which is affiliated or legally linked to a participant. The entity carries out work under the conditions laid down in the Grant Agreement, supplies goods or provides services for the action, but did not sign the Grant Agreement. A third party abides by the rules applicable to its related participant under the Grant Agreement with regard to eligibility of costs and control of expenditure. BCAM - BASQUE CENTER FOR APPLIED MATHEMATICS Spain Net EU contribution € 768 750,40 Address Al mazarredo 14 48009 Bilbao See on map Region Noreste País Vasco Bizkaia Activity type Research Organisations Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 768 750,40