Collaborative Research: New Decouplings and Applications

主要研究人员最近提出了一套新的工具，称为解耦，可以成功地量化波在不同方向上传播的相互作用方式。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要突破。更准确地说，丢番图方程是包含整数的潜在复杂方程组，数学家对计算这些方程组的解的数量很感兴趣。与波不同，数字不会振荡，至少不会以一种明显的方式振荡，但人们可以将数字视为频率，从而将它们与波联系起来。这样，与计算丢番图系统的解的数目有关的问题可以用量化波干涉的语言来重新表述。该项目将进一步扩展解耦的范围，以解决谐波分析和数论中的基本问题。该项目将为数学界的大部分人提供可访问和有用的新工具。解耦在解决数论、偏微分方程和谐波分析等不同领域的各种问题方面被证明是非常灵活的。一个有待解决的重要问题是关于小空间球上曲线的解耦不等式。此外，圆锥体带来了许多有趣的问题，即使在三维空间中也是如此。平方函数估计，局部平滑猜想和解耦成小帽只是锥体相关问题的几个例子。在这些问题上取得进展可能会导致在许多其他问题上取得进展。最后，解耦和多项式方法的结合最近在弯曲流形的约束理论方面取得了重大进展。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。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.