EBIPHA

Objective:

This proposal outlines a four-year research plan on Inverse Problems (IPs) and Harmonic Analysis (HA), exploiting shared tools and techniques in order to address research questions that will advance our knowledge in these areas. The team comprises two Principal Investigators, Ikerbasque Research Associate Professors Pedro Caro (BCAM) and Ioannis Parissis (UPV/EHU), supported by Aingeru Fernández-Bertolin and Andoni García (both UPV/EHU). Several PhD students and postdocs are included in the working group. Pedro Caro focuses on inverse problems (IPs) arising in partial differential equations (PDEs), particularly in the regularity theory and minimal assumptions. He is currently supervising one PhD student in IPs and has substantial experience in mentoring young researchers and a strong record in IPs and the theory of PDEs. Ioannis Parissis is responsible for the harmonic analysis (HA) part of the project, with a focus on the interaction between thin sets and Fourier multiplier operators. He is also involved in the study of inverse problems related to quantum Hamiltonians. Parissis is currently supervising a PhD student working on related topics and has a proven research and training record in Euclidean HA. Aingeru Fernández-Bertolin works on the intersection of HA and PDEs, specifically unique continuation principles as dynamical versions of uncertainty principles. Andoni García studies uniqueness in the inverse Calderón problem for anisotropic media and quantum scattering in rough media.. The group has a proven track record of mentoring PhD students and postdoctoral researchers who have progressed to academic and research positions internationally. We are also applying for funding for one PhD student and one postdoctoral researcher who will support the outlined objectives. Inverse problems in PDEs aim to determine the coefficients of equations or systems from data, modeling applications such as non-destructive testing and medical imaging. This proposal addresses two important subareas: Inverse Boundary Value Problems: Determining the physical properties of a medium through boundary data. Initial-to-Final Inverse Problems: A novel framework to determine a time-dependent partial differential operator from the initial and final values of its solutions. The research focuses on identifying the necessary assumptions about coefficients' nature (e.g., scalar, vector) and regularity (e.g., integrability, differentiability) to enable reconstruction. The HA component focuses on thin sets with small additive structure, such as lacunary sequences. The suggested research focuses on how thinness relates to the boundedness of Fourier multiplier operators with frequency singularities on such sets. Directional Singular Integrals: These are Fourier multiplier operators with frequency singularities along thin sets of directions. They are intimately connected with important problems in HA such as the maximal Nikodym and Kakeya conjectures. In this proposal we aim at quantifying the correspondence between thin sets and the mapping properties of multiparameter Fourier multipliers. This proposal seeks to advance the current knowledge in IPs and HA. The teams expertise, prior collaborative success, and focus on mentoring early- career researchers highlights the potential impact of this research project.