Geometric Harmonic Analysis: Affine and Frobenius-Hörmander Geometry

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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. There are many basic theoretical open questions in this area of mathematics which remain unsolved despite the many incredible successes the field has already achieved. In this project a family of questions will be studied in the area of geometric averages. These 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). The main goals of this project will advance a number of related areas of mathematics and may influence future imaging technologies. Graduate students are involved in the project.PI will focus on several topics in mathematical analysis related to the development of new geometric methods for a family of questions relating to the mapping properties of Radon-like operators, oscillatory integrals, and Fourier restriction operators. The specific classes of operators to be studied include multilinear Radon-like averaging operators as well as related nonsingular oscillatory questions of the sort first studied by other researchers. 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. PI will use a geometric and combinatorial approach as the main toolkit, which includes a variety of new tools developed within the last 5 years incorporating techniques from geometric invariant theory, geometric measure theory, decoupling theory, and other areas.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.