Final Report Summary - TMHA (Transversal Multilinear Harmonic Analysis)
While harmonic analysis has long benefited from considerable interconnectivity within mathematics and the physical sciences, research in recent years has taken these links to an unexpected level both in terms of depth and variety. In particular, the development of multilinear harmonic analysis over the last 20 years has shed new light on our understanding of fundamental properties of the ubiquitous Fourier transform, and more general oscillatory integral operators, providing a framework within which unexpected and highly inventive methodologies may be brought to bear on resilient problems at the heart of the subject. The main aim of this project was to develop this multilinear theme in the context of problems related to notions of curvature and transversality of smooth manifolds, with particular focus on oscillatory and Radon-like integral transforms and their relationships with geometrically-defined (or Kakeya-type) maximal inequalities. This has led to a variety of outcomes that deepen our understanding of the explicitly geometric aspects of modern harmonic analysis. Of particular significance has been the establishment of a range of optimal, or near-optimal, variants of the celebrated Brascamp-Lieb inequality - a very general and far-reaching classical functional inequality in modern analysis, celebrated for its numerous applications and connections across mathematics. This has culminated in the establishment of a nonlinear variant of this inequality, providing bounds on a broad class of quite general positive multilinear forms, with substantial applications in harmonic analysis and partial differential equations. In addition to direct applications of these results, the methods developed in the project serve as an encouraging illustration of the scope and effectiveness of recursive multi-scale approaches to harmonic and geometric analysis. A key intermediate success of the project was the establishment of a regularity theory for the so-called Brascamp-Lieb constant, which combined with recent developments in the field, yields a family of Fourier restriction and Kakeya-type inequalities of striking breadth. Instances of these inequalities have since proved to be key ingredients in the development of "decoupling theory" (also known as Wolff inequalities), that has been extraordinarily effective across mathematics over the last five years. The project also succeeded in establishing a framework for controlling broad classes of highly oscillatory integral operators solely in terms of quite transparent geometric operations such as averaging over balls and rectangles, in the spirit of an influential conjecture of Stein. These classes of oscillatory integrals include solutions of a variety of partial differential equations from the mathematics of wave propagation, leading to purely geometric interpretations of dispersive effects which function at all scales of space and time. The project also developed novel perspectives on the theory of space-time bounds for solutions to such equations, including the classical Schroedinger, wave and kinetic transport equations, capitalising on its distinct multilinear perspective.