Topics in Real and Discrete Harmonic Analysis

This project focuses on harmonic analysis, a branch of mathematics that develops tools for decomposing functions and signals into simpler components. Theoretical ideas and methods from harmonic analysis play an important role in areas such as signal processing, medical imaging, fluid mechanics, data analysis and number theory. Foundational research in harmonic analysis enlarges the mathematical toolbox available to these and other fields, and it often reveals unexpected connections between different parts of mathematics. The project studies several central operators that average, filter or detect structure in functions. One theme of this research concerns how the behavior of such operators depends on the geometry of the sets of scales on which they act, including sets with fractal structure. Another theme concerns operators that measure interactions among several functions at once, connecting to questions about patterns in large sets and long-term behavior in dynamical systems. In summary, the project furthers the progress of science through an improved theoretical understanding of these operators. The project also includes training and mentoring of students and early-career researchers in mathematics, organization of seminars and scientific meetings, and curriculum development. The technical focus of the project is on problems in real and discrete harmonic analysis. One major research direction concerns geometric maximal functions associated with restricted and fractal sets of dilations. More specifically, the project addresses sharp mapping properties of spherical maximal operators between Lebesgue spaces and their dependence on notions of fractal dimension such as the Assouad spectrum, quasi-Assouad dimension and the Legendre-Assouad function. Closely related work concerns fractal variants of local smoothing estimates for the wave equation with connections to wave packet decompositions, square functions and Fourier restriction theory. Another main direction concerns multilinear singular integrals such as multilinear versions of the Hilbert transform. A central component is the study of multilinear smoothing inequalities, which are closely tied to time-frequency analysis, oscillatory integral estimates and ideas from additive combinatorics. Such inequalities have applications to polynomial patterns in dense sets and pointwise convergence questions in ergodic theory. Additional work includes analysis on the Hamming cube, discrete analogues, Carleson-type operators, and has connections with artificial intelligence (specifically, machine learning) through formalization of mathematics in Lean. 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.