Small Cap and Large Cap Decoupling

小盘股和大盘股脱钩

基本信息

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

项目摘要

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.

在科学和应用的许多分支中，朝着不同方向相互作用的波浪在不同方向相互作用的程度很重要。在过去的十年中，PI开发了一组新的数学工具，称为脱钩，以衡量这种相互作用的程度。尽管这些工具最初是针对有关微分方程的某些问题的，但它们也导致了数量理论的重要突破。更确切地说，双方方程是多项式方程的系统，涉及具有潜在复杂解决方案的整数。数学家有兴趣计算此类系统的解决方案的数量。与波浪不同，数字不会振荡，至少不是以明显的方式振荡，而是可以将数字视为频率，从而将它们与波浪联系起来。通过这种方式，可以用量化波浪干扰的语言来重新塑造与计算二芬太丁系统解决方案的数量有关的问题。该项目将进一步将脱钩的范围扩展到谐波分析和数理论中基本问题的范围。将开发新的工具，这些工具将对数学社区的大部分地区访问且有用。此外，首席研究人员将组织暑期学校和研讨会，这些讲习班将教育其他数学领域的年轻研究人员和专家，涉及十字形的适用性。事实证明，Decouplings在数字理论，部分差分方程和和谐分析等各种领域的广泛问题中已成功地解决了成功。通过该项目，计划了这些方法在新方向上的适用性。待解决的一个重要问题圈子涉及小空间球上曲线的脱钩不等式。将进一步寻求对称为“大帽脱钩的大帽脱钩”的新现象的调查。大型盖脱钩涉及平方根取消歧管的更粗糙分区。该项目的第三部分致力于调查Tori的子手法的限制。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，认为值得通过评估值得支持。