Geometric Harmonic Analysis: Affine and Frobenius-Hormander Geometry for Multilinear Operators

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The mathematics of geometric averages, also known as Radon-like operators, is of fundamental importance in a host of technological applications related to imaging: CT, SPECT, and NMR, as well as RADAR and SONAR applications, all depend on a deep understanding of the Radon transform, and related ideas appear in optical-acoustic tomography, scattering theory, and even some motion-detection algorithms. Somewhat surprisingly, there are many basic theoretical problems in this area of mathematics which remain unsolved despite the many incredible successes the field has already achieved. This project proposes to study a family of questions in the area of geometric averages which, roughly speaking, correspond to quantifying the relationship between small changes in the imaged objects and the expected changes in measured data (which in practice is processed computationally to recover an approximate picture of the original object). Achieving the main goals of this project would lead to advances in a number of related areas of mathematics and may influence future imaging technologies.More precisely, the project will focus on several topics in mathematical analysis related to the development of new geometric methods for multilinear operators, with specific emphasis on the establishment of uniform estimates. The specific classes of operators to be studied include multilinear Radon-like averaging operators as well as related nonsingular oscillatory problems of the sort first studied by Christ, Li, Tao, and Thiele. Major special cases deserving mention include multiparameter sublevel set estimates, maximal curvature for Radon-like transforms of intermediate dimension, degenerate Radon transforms in low codimension, Fourier restriction and related generalized determinant functionals, Phong-Stein operator van der Corput methods, and multilinear oscillatory integrals of convolution and related types. A secondary goal of the project is to apply these ideas to PDEs. A geometric and combinatorial approach will be used as the main toolkit since these methods have a consistent record of success, and the principal investigator has, in particular, made major progress in this direction in recent years. These methods incorporate ideas from a wide variety of areas in mathematics, including Geometric Invariant Theory, Geometric Measure Theory, Convex Geometry, and Signal Processing.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.