Física, Química y Matemáticas (FQM) Universidad del País Vasco The project covers a wide scope ranging from theoretical aspects of Fourier or Harmonic Analysis (Uncertainty principle, Singular Integrals, Kakeya sets) to Geometric Measure Theory (Harmonic Measure) or Quantum Mechanics (Many-body system, Dirac equation) and passing from Analysis of Partial of Differential Equations (Control theory, Unique continuation) as well as more applied aspects like Numerical Analysis of problems from fluid mechanics. Being more precise we propose to study the following topics: (A) Mathematical Physics 1) General theory of weakly interacting fermions. 2) Dirac Equation 3) Non self-adjoint Hamiltonians 4) Kinetic equations 5) Harmonic Analysis and Quantum Mechanics (B) Analysis of Partial Differential Equations 1) Unique continuation properties related to Uncertainty Principles 2) Control Theory 3) Inverse problems 4) The Vortex Filament Equation 5) Numerical Analysis (C) Harmonic Analysis 1) Singular and maximal averages associated with Kakeya and Nikodym sets. 2) Spherical maximal functions 3) Extrapolation theory, classical operators, Lorentz spaces and multiparameter Harmonic Analysis. 4) Parabolic rectifiability and caloric measure 5) Degenerate Fractional Poincare-Sobolev inequalities and Harmonic Analysis