Flow methods in geometric aspects of harmonic analysis.

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This is a project in Euclidean Harmonic Analysis focussing on the application of flow methods in the context of multilinear functional inequalities. The applicability of such methods in harmonic analysis has seen an unexpected surge over the last 5-10 years, with important consequences in areas such as partial differential equations and number theory. The purpose of this project is to refine these methods in various contexts related to the multilinear restriction theory of the Fourier transform. More specifically, the project aims to obtain near-monotonicity statements for the so-called nonlinear Brascamp-Lieb functional, and explore its consequences for the nonlinear Brascamp-Lieb inequality.