Small Cap and Large Cap Decoupling

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The extent to which waves traveling in different directions interact with each other is important in many branches of science and applications. Over the last decade the PI has developed a new set of mathematical tools called decouplings to measure this extent of interaction. While these tools were initially intended for certain questions about differential equations, they have also led to important breakthroughs in number theory. More precisely, Diophantine equations are systems of polynomial equations involving whole numbers with potentially complicated solutions. 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, questions related to counting the number of solutions to Diophantine systems can be rephrased in the language of quantifying wave interferences. The project will further extend the scope of decouplings towards the resolution of fundamental problems in harmonic analysis and number theory. New tools will be developed that will be accessible and useful to a large part of the mathematical community. In addition, the Principal Investigator will organize summer schools and workshops that will educate both young researchers and experts from other areas of mathematics about the applicability of decouplings.Decouplings have proved successful in addressing a wide range of problems in such diverse areas as number theory, partial differential equations and harmonic analysis. Through the project a further expansion is planned of the applicability of these methods in new directions. One important circle of questions that remain to be addressed concerns the decoupling inequalities for curves on small spatial balls. An investigation of a new phenomenon called large cap decoupling will further be sought after. Large cap decoupling is concerned with square root cancellation for coarser partitions of manifolds. A third part of the project is devoted to investigating restrictions of exponential sums to submanifolds of tori.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.