Collaborative Research: New Decouplings and Applications

合作研究：新的解耦和应用

• 批准号: 1800305
• 负责人: Ciprian Demeter
• 金额: $ 27万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800305&HistoricalAwards=false
• 关键词: Collaborative; Research; New; Decouplings; Applications

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.

主要研究人员最近提出了一套称为解耦的新工具，可以成功量化沿不同方向传播的波相互作用的方式。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，丢番图方程是涉及整数的潜在复杂方程组，数学家对计算此类系统的解的数量感兴趣。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。这样，与计算丢番图系统解的数量相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。该项目将使数学界的大部分人都能使用和使用新工具。事实证明，解耦在解决数论、偏微分方程和调和分析等不同领域的各种问题方面非常灵活。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。此外，圆锥体也带来了很多有趣的问题，即使是在三维空间中也是如此。平方函数估计、局部平滑猜想以及解耦为小型大写字母只是锥体相关问题的几个例子。这些问题上的进展可能会导致许多其他问题上的进展。最后，解耦和多项式方法的结合最近在弯曲流形的限制理论中取得了重大进展。主要研究人员打算在这个令人兴奋且快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

Decouplings for real analytic surfaces of revolution

真实旋转解析曲面的解耦

• 发表时间: 2020
• 期刊: Lecture notes in mathematics
• 作者: Bourgain, Jean;Demeter, Ciprian;Kemp, Dominique
• 通讯作者: Kemp, Dominique

Sharp $$\ell ^p$$-Improving Estimates for the Discrete Paraboloid

Sharp $$ell ^p$$ - 改进离散抛物面的估计

• DOI: 10.1007/s00041-020-09801-2
• 发表时间: 2021
• 期刊: Journal of Fourier Analysis and Applications
• 影响因子: 1.2
• 作者: Dasu, Shival;Demeter, Ciprian;Langowski, Bartosz
• 通讯作者: Langowski, Bartosz

Small cap decouplings

小盘股脱钩

• DOI: 10.1007/s00039-020-00541-5
• 发表时间: 2020
• 期刊: Geometric and Functional Analysis
• 影响因子: 2.2
• 作者: Demeter, Ciprian;Guth, Larry;Wang, Hong
• 通讯作者: Wang, Hong

Three applications of the Siegel mass formula

西格尔质量公式的三种应用

• 发表时间: 2020
• 期刊: Lecture notes in mathematics
• 作者: Bourgain, Jean;Demeter, Ciprian
• 通讯作者: Demeter, Ciprian