BCAM SO Course

Dispersive waves. Linear Schrödinger equation. Fundamental solution. Time decay. Non endpoint Strichartz estimates.

Endpoint Strichartz estimates (Keel-Tao).

H^1-subcritical NLS: Local Cauchy Theory by compactness arguments. Global Cauchy theory in the defocusing case.

Focusing H^1-subcritical NLS: Global Cauchy Theory. Gagliardo-Nirenberg inequalities, Virial Theorem. Blow-up á la Glassey. 

Scattering for NLS. Existence and asymptotic completeness of Wave Operators. Morawetz estimates. Interaction Morawetz estimates. Pseudoconformal transformation.

The Kenig-Merle Strategy.

The Kenig-Merle Strategy.

The Kenig-Merle Strategy.

The Kenig-Merle Strategy.

Linear Schrödinger equation with potentials. Subcritical potentials, time-decay and Strichartz estimates. 

Scaling critical potentials: Strichartz estimates and time-decay.

Uncertainty principles 1: Hardy inequalities and applications.

Uncertainty principles 2: dynamical uncertainty.

Spectral stability. Embedded eigenvalues. Mourre Theory, Birman-Schwinger Principle, method of multipliers. Eigenvalue localization (Sobolev inequalities, Birman-Schwinger).

Since January 2023, I have been an Ikerbasque and Ramón y Cajal Research Fellow at UPV/EHU in Bilbao. Prior to this, I worked as a Postdoctoral Researcher at various institutions in Germany, the Czech Republic, France, and Spain. I earned my PhD in Rome in 2017.

My research is concerned with the investigation of characterizing features connected with dispersive partial differential equations, the mathematical rigorous study of spectral properties of self-adjoint/non-self-adjoint Hamiltonians of quantum mechanics, and Hardy-type inequalities.

Luca Fanelli (Bari, Italia, 1979) PhD in Piano (Rotterdams Conservatorium 2002) and Mathematics (University of Roma “La Sapienza” 2008). His research activity is about Spectral Theory, Fourier Analysis and PDE. He has been PI of several research projects, and he has been awarded 3 times the Teaching Prize at the University of Roma “La Sapienza”, where he has been Associate Professor in Math. Analysis until 2020. At the moment, he is Ikerbasque Research Professor at the Math. Department of UPV/EHU and at BCAM, member of the Executive Committee of SEMA (Sociedad Española de Matemática Aplicada) and coordinator of the section “Innovación” at the Academy of Medical Sciences  of Bilbao. In 2023, he created the program “2M: Mathematics & Music”, in which he organizes divulgation activities, concerts, and the Music Competition “Villa de Bilbao”, celebrating its second edition in 2025.