Small Cap and Large Cap Decoupling

在科学和应用的许多分支中，不同方向传播的波相互作用的程度是重要的。在过去的十年里，PI开发了一套新的数学工具，称为解耦，以衡量这种相互作用的程度。虽然这些工具最初是为了解决有关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，丢番图方程是涉及具有潜在复杂解的整数的多项式方程组。数学家对计算这类系统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扰的语言来重新表述。该项目将进一步扩大解耦的范围，以解决调和分析和数论中的基本问题。将开发新的工具，这些工具将对数学界的很大一部分人可用和有用。此外，首席调查员将举办暑期班和工作坊，教育年轻研究人员和其他数学领域的专家了解解耦的适用性。解耦已被证明成功地解决了数论、偏微分方程式和调和分析等不同领域的广泛问题。通过该项目，计划进一步扩大这些方法在新方向上的适用性。一个有待解决的重要问题是关于小空间球上的曲线的去耦合不等。对一种名为大盘股脱钩的新现象的调查将进一步受到追捧。大帽解耦与较粗的流形划分的平方根抵消有关。该项目的第三部分致力于调查指数和对托里子流形的限制。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.