Tomographic Fourier Analysis

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This project lies in the field of euclidean harmonic analysis, and in particular the so-called restriction theory of the Fourier transform. This mathematical theory concerns the manner in which families of waves propagating in different directions in euclidean space can interact, and establishes deep inequalities that estimate this interaction effectively. This area has seen remarkable growth and impact over the last decade, considerably deepening its connections with other branches of mathematics, such as differential equations, combinatorial geometry, algebraic geometry and number theory. This project places particular emphasis on the development of a powerful and widely-applicable new methodology, and its potential to transform Fourier restriction theory. This methodology, naturally termed Tomographic Fourier Analysis, is designed to reveal the extent to which superpositions of waves in space (referred to as Fourier extensions) may be studied effectively via their "sections" or "slices". This simple idea opens a new and direct route by which classical methods of Fourier analysis may be applied to contemporary problems in harmonic analysis. The specific objectives are to establish a range of important conjectural inequalities that control Fourier extensions in terms of classical tomographic transforms, such as the X-ray and Radon transforms (the so-called Mizohata-Takeuchi and Stein conjectures). In particular, establishing such control would profoundly strengthen the longstanding high-profile interface of harmonic analysis with combinatorial geometry, and clarify the mysterious relationship between the celebrated Fourier restriction and Kakeya conjectures.Complex wave-like phenomena of the type described above pervade the mathematical sciences, and are notoriously difficult to understand. The tools developed in the proposed research allow the underlying oscillatory structures (so-called oscillatory integral operators) to be viewed in purely geometric and combinatorial ways. This has the potential for significant applications and benefits in the longer term. Furthermore, the methodology (Tomographic Fourier Analysis), as its name indicates, has the potential to benefit mathematics through novel two-way interactions between the harmonic analysis and inverse problems communities.