CAREER: Decoupling Theory, Oscillatory Integral Theory, and Their Applications in Analytic Number Theory and Combinatorics

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This project concerns research in harmonic analysis and analytic number theory. Harmonic analysis studies how general functions can be represented by sums of simpler functions, for instance, trigonometric functions. Analytic number theory is a branch of number theory that uses tools from mathematical analysis to answer questions concerning the integers. Recently, tools in harmonic analysis have proven to be very efficient in answering questions originating from analytic number theory. This project continues the investigation in the intersection of these two fields, in the hope of creating a dictionary between them that would allow researchers in one field to translate their tools to the other. The project also provides undergraduate students access to the forefront of current research. The PI plans to organize summer schools to attract more undergraduate and early graduate students to carry out interdisciplinary research in harmonic analysis and analytic number theory.The project involves work in decoupling theory, oscillatory integral theory, and their applications in analytic number theory and combinatorics. One goal in the development of decoupling theory is to obtain sharp decoupling inequalities for all polynomial surfaces that are translation-dilation invariant (TDI in short). This would imply sharp bounds on the number of integral solutions to all TDI systems of Diophantine equations. As a first step towards this goal, the PI will study two particularly important systems: TDI systems of monomials and TDI systems generated by one polynomial. In oscillatory integral theory, the PI also will study two problems. The first problem asks for the optimal Sobolev regularity estimates for an averaging operator along moment curves. The second problem asks for sharp Lebesgue estimates for maximal averaging operators along moment curves. It is expected that local smoothing estimates for linear wave equations and related decoupling inequalities will play key roles in the study of these two problems.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.