Small Cap and Large Cap Decoupling

小盘股和大盘股脱钩

• 批准号: 2055156
• 负责人: Ciprian Demeter
• 金额: $ 30.4万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2055156&HistoricalAwards=false
• 关键词: Small; Cap; Large; Decoupling

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.

在不同方向传播的波相互作用的程度在科学和应用的许多分支中都很重要。在过去的十年中，PI 开发了一套称为解耦的新数学工具来衡量这种相互作用的程度。虽然这些工具最初是用于解决微分方程的某些问题，但它们也带来了数论的重要突破。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。数学家有兴趣计算此类系统的解决方案的数量。 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。 这样，与计算丢番图系统解的数量有关的问题就可以用量化波干扰的语言来重新表述。该项目将进一步扩展解耦的范围，以解决调和分析和数论中的基本问题。新的工具将会被开发出来，对数学界的大部分人来说都是可以使用和有用的。此外，首席研究员还将组织暑期学校和研讨会，对来自其他数学领域的年轻研究人员和专家进行解耦的适用性教育。事实证明，解耦可以成功地解决数论、偏微分方程和调和分析等不同领域的各种问题。通过该项目，计划进一步扩展这些方法在新方向的适用性。有待解决的一系列重要问题涉及小空间球上曲线的解耦不等式。人们将进一步寻求对一种称为大盘脱钩的新现象的调查。大帽解耦涉及流形较粗分区的平方根消除。该项目的第三部分致力于研究环面子流形的指数和限制。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。

项目成果

On restriction of exponential sums to hypersurfaces with zero curvature

关于零曲率超曲面的指数和限制

• DOI: 10.1007/s00208-022-02361-4
• 发表时间: 2023
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Demeter, Ciprian