FRG: Collaborative Research: New Trends in Harmonic Analysis

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Abstract for 0456538 Lacey, 0456306 Iosevich, and 0456490 MagyarThe proposed activity will focus a set of topics of current interest in Harmonic Analysis: the Kakeya conjecture, the Fourier restriction problem and Carleman estimates, the smoothing conjecture for the wave equation, and Zygmund's conjecture on the differentiation along Lipschitz families of lines. Equally central are newer questions that are being identified and studied, including the discrete analogues of classical operators, aspects of the Fuglede conjecture, Fourier integral operators with degenerate canonical relations and new classes of singular integrals in complex analysis. Past advances in these subjects have drawn influences from subjects such as Combinatorics, Additive Combinatorics, and Number Theory to name some prominent areas. In turn, these questions are making contributions to these same areas. The coordination of the efforts that this proposal will permit should accelerate advances on this broad range of topics. The proposed activities concern central questions that will result in new modes of analytical technique that bear on questions of, for instance, behavior of waves in higher dimensions, and different aspects of the subtle distinctions between discrete, i.e. digital, and continuous objects. In addition, the projects will draw upon methods and techniques from a range of different areas of mathematics. The breadth and sophistication of the analytical methods in the subject sheds new light on the interrelationships between these areas. It also can be an obstacle to continued research. This project has as an important of focus the training of graduate students and postdocs in these emerging areas of research, and the wide variety of techniques used in their analysis. These efforts will foster a next generation of mathematicians, critical to the nations scientific infrastructure.