Collaborative Research: New Decouplings and Applications

主要研究人员最近提出了一套新的工具，称为“叠加”，可以成功地量化不同方向传播的波相互作用的方式。虽然这些工具最初是为了解决微分方程的某些问题，但它们也导致了数论的重大突破。更确切地说，丢番图方程是一个潜在的复杂的方程组，涉及整数，数学家们对计算这样的方程组的解的个数很感兴趣。与波不同，数字不会振荡，至少不会以明显的方式振荡，但人们可以将数字视为频率，从而将它们与波联系起来。通过这种方式，与计算丢番图系统的解的数量有关的问题可以用量化波干扰的语言来重新表述。这个项目将进一步扩大对调和分析和数论的基本问题的解决方案的范围。该项目将使新的工具，方便和有用的大部分数学界。解耦已被证明是非常灵活的，在解决各种各样的问题，在这样的不同领域的数论，偏微分方程和谐波分析。一个重要的圈的问题，仍然有待解决的关注的解耦不等式的曲线上的小空间球。此外，圆锥体提出了很多有趣的问题，即使是在三维空间中。平方函数估计，局部平滑猜想和解耦成小帽只是锥相关问题的几个例子。在这些问题上取得的进展有可能导致在许多其他问题上取得进展。最后，最近，组合的多项式和多项式方法导致显着进展的限制理论的曲流形。主要研究人员打算在这个令人兴奋和迅速发展的领域寻求进一步的改进。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.