This fellowship will enable the Experienced Researcher Dr. Gian Maria Dall'Ara and Dr. Alessio Martini, as Host Researcher based at the University of Birmingham, to carry out innovative research at the interface of harmonic and complex analysis. This very active research area has deep connections with the most disparate fields, from algebraic, complex and subriemannian geometry to analysis on Lie groups and numerical harmonic analysis. Despite the spectacular developments of the last decades, many fundamental problems remain open and several phenomena lack a conceptual explanation. Dall'Ara will bring a point of view influenced by mathematical physics in which some of the key questions are interpreted in terms of uncertainty principles for generalized Schrödinger operators, and he has already proved the effectiveness of this approach in a number of situations. Dr. Alessio Martini's mastery of harmonic analysis in various non-Euclidean settings will provide the crucial ingredient to disclose the full potential of these novel ideas. The project consists of various interconnected working packages, mainly focusing on two mathematical objects naturally attached to real hypersurfaces embedded in complex manifolds: Cauchy-Szegö projections and spectral multipliers of Kohn Laplacians. The former are higher dimensional incarnations of the classical Cauchy integral, and as such they are of central importance in modern complex analysis in several variables. The project will increase our understanding of the nature of their singularities and mapping properties under general geometric assumptions involving for example the Ricci curvature. Kohn Laplacians are the natural Laplacians in this context and the study of their spectral multipliers fits into a wider set of problems lying at the heart of contemporary harmonic analysis. Our methods are a combination of geometric analysis, singular integral theory and mathematical physics.
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