Collaborative Research: New Decouplings and Applications

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The principal investigators have recently advanced a new set of tools called decouplings, that can successfully quantify the ways in which waves traveling in different directions interact with each other. While these tools were initially intended for certain problems about differential equations, they have also led to important breakthroughs in number theory. More precisely, Diophantine equations are potentially complicated systems of equations involving whole numbers, and mathematicians are interested in counting the number of solutions to such systems. Unlike waves, numbers do not oscillate, at least not in an obvious manner, but one can think of numbers as frequencies, and thus associate them to waves. In this way, problems related to counting the number of solutions to Diophantine systems can be rephrased in the language of quantifying wave interferences. This project will further extend the scope of decouplings towards the resolution of fundamental problems in harmonic analysis and number theory. The project will make new tools accessible and useful to a large part of the mathematical community.Decouplings have proved remarkably flexible in addressing a wide variety of problems in such diverse fields as number theory, partial differential equations and harmonic analysis. One important circle of questions that remain to be addressed concerns the decoupling inequalities for curves on small spatial balls. Also, the cone poses a lot of interesting problems, even in three dimensions. The square function estimate, the local smoothing conjecture and, the decoupling into small caps are just a few examples of related problems for the cone. Progress on these problems is likely to lead to progress on many other problems. Finally, the combination of decouplings and the polynomial method has recently led to significant progress in the restriction theory of curved manifolds. The principal investigators intend to seek further improvements in this exciting and rapidly developing area.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.