Some problems at the interface of harmonic analysis, number theory, and combinatorics
调和分析、数论和组合学接口的一些问题
基本信息
- 批准号:1600840
- 负责人:
- 金额:$ 16.44万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:2016
- 资助国家:美国
- 起止时间:2016-07-01 至 2020-06-30
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
So-called Ramsey theory deals with the problem of finding structures in large but otherwise disorganized sets. In the geometric setting it is to show that such sets contain a translated and rotated copy of a given finite set, or of its sufficiently large dilates. In other words it is to study the occurrence of geometric patterns. Over the past fifteen years there has been a remarkable progress of the study of linear patterns, developing and introducing tools from mathematical analysis, often referred to as higher-order Fourier analysis. Among the major achievements is the celebrated result of Green and Tao, which states that there are arbitrary long sequences of equally spaced prime numbers. This project builds on this development, and one of its major objectives is to develop analytic tools to understand the occurrence of geometric and arithmetic (i.e., defined by equations) structures in large but otherwise arbitrary sets. The problems arise in the context of the prime and integer lattice and also in classical Euclidean spaces. The principal investigator's approaches involve the interplay of techniques from discrete harmonic analysis and number theory, in addition to a new ingredient, ideas from additive combinatorics.The first motivational context for the project is that of prime numbers: to study nonlinear relations among the primes and to investigate the related problem of finding geometric constellations among points with prime coordinates. The underlying philosophy of considering the primes as a random subset of the integers leads naturally to the study of analogous questions in large sets of integer points and also in large measurable subsets of Euclidean spaces. Geometric structures in such sets are not well understood. The project aims to develop a general approach based on the modern point of view of additive combinatorics; namely, to establish appropriate notions of randomness that control the frequency at which a certain pattern occurs and to prove structure theorems for sets that are not suitably random. The underlying constructs are analytic and are related to objects studied in discrete harmonic analysis such as maximal operators and Radon transforms acting on functions defined on the integer lattice. Finally, the project aims to study geometric patterns in large measurable subsets of Euclidean spaces from this novel point of view, strengthening the connections between additive combinatorics and classical harmonic analysis.
所谓的拉姆西理论涉及在大型但混乱的集合中找到结构的问题。在几何环境中,它表明此类集合包含给定有限集的翻译和旋转副本,或者其足够大的扩张。换句话说,它是研究几何模式的发生。在过去的十五年中,线性模式的研究取得了显着的进展,从数学分析中开发和介绍了工具,通常称为高阶傅立叶分析。主要成就是格林和道的著名结果,该结果指出有任意的长序列相同的质数。该项目建立在此开发的基础上,其主要目标之一是开发分析工具,以了解几何和算术的发生(即由方程定义)结构以大而任意的集合。问题是在素数和整数晶格的背景下以及经典的欧几里得空间中出现的。首席研究者的方法涉及来自离散谐波分析和数量理论的技术相互作用,除了新成分,来自添加剂Compinatorics的想法。将素数作为整数的随机子集的基本哲学自然导致研究大量整数点以及在欧几里得空间的大量可测量子集中的类似问题。这种集合中的几何结构尚不清楚。 该项目旨在根据添加剂组合学的现代角度开发一种一般方法;也就是说,建立适当的随机性概念,以控制发生特定模式的频率并证明无法正常随机的集合的结构定理。基础构建体是分析性的,并且与离散谐波分析中研究的对象有关,例如最大算子和ra ravers转换作用于整数晶格上定义的功能的对象。最后,该项目旨在从这个新颖的角度研究欧几里得空间的大量可测量子集中的几何模式,从而增强添加剂组合学和经典谐波分析之间的联系。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
数据更新时间:{{ journalArticles.updateTime }}
{{
item.title }}
{{ item.translation_title }}
- DOI:
{{ item.doi }} - 发表时间:
{{ item.publish_year }} - 期刊:
- 影响因子:{{ item.factor }}
- 作者:
{{ item.authors }} - 通讯作者:
{{ item.author }}
数据更新时间:{{ journalArticles.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ monograph.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ sciAawards.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ conferencePapers.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ patent.updateTime }}
Akos Magyar其他文献
Akos Magyar的其他文献
{{
item.title }}
{{ item.translation_title }}
- DOI:
{{ item.doi }} - 发表时间:
{{ item.publish_year }} - 期刊:
- 影响因子:{{ item.factor }}
- 作者:
{{ item.authors }} - 通讯作者:
{{ item.author }}
{{ truncateString('Akos Magyar', 18)}}的其他基金
Discrete problems in harmonic analysis with applications to ergodic theory and additive number theory
调和分析中的离散问题及其在遍历理论和加性数论中的应用
- 批准号:
0803190 - 财政年份:2008
- 资助金额:
$ 16.44万 - 项目类别:
Continuing Grant
FRG: Collaborative Research: New Trends in Harmonic Analysis
FRG:协作研究:谐波分析的新趋势
- 批准号:
0456490 - 财政年份:2005
- 资助金额:
$ 16.44万 - 项目类别:
Standard Grant
Discrete Problems in Harmonic Analysis, Ergodic Theorems and Singularities
调和分析、遍历定理和奇点中的离散问题
- 批准号:
0202021 - 财政年份:2002
- 资助金额:
$ 16.44万 - 项目类别:
Continuing Grant
Problems in Analysis Related to Lattice Points and Singularities
与格点和奇点相关的分析问题
- 批准号:
9970899 - 财政年份:1999
- 资助金额:
$ 16.44万 - 项目类别:
Standard Grant
相似国自然基金
Stokes界面问题非拟合压力鲁棒数值方法与理论分析
- 批准号:12301469
- 批准年份:2023
- 资助金额:30 万元
- 项目类别:青年科学基金项目
两类界面问题的新颖高精度数值方法及应用
- 批准号:12371418
- 批准年份:2023
- 资助金额:44.00 万元
- 项目类别:面上项目
低温异质集成硅基片上光源的界面机理与光耦合关键问题研究
- 批准号:62304251
- 批准年份:2023
- 资助金额:30.00 万元
- 项目类别:青年科学基金项目
界面问题的高效虚拟有限元法
- 批准号:12371397
- 批准年份:2023
- 资助金额:43.5 万元
- 项目类别:面上项目
求解界面问题的扩展虚拟元方法
- 批准号:12371369
- 批准年份:2023
- 资助金额:43.5 万元
- 项目类别:面上项目
相似海外基金
Canonical mean curvature flow and its application to evolution problems
正则平均曲率流及其在演化问题中的应用
- 批准号:
23H00085 - 财政年份:2023
- 资助金额:
$ 16.44万 - 项目类别:
Grant-in-Aid for Scientific Research (A)
Spectral properties of interface problems for Maxwell systems
麦克斯韦系统界面问题的谱特性
- 批准号:
EP/W007037/1 - 财政年份:2022
- 资助金额:
$ 16.44万 - 项目类别:
Research Grant
確率インバージョンによる地中内部の可視化
使用随机反演可视化地下内部
- 批准号:
22K20601 - 财政年份:2022
- 资助金额:
$ 16.44万 - 项目类别:
Grant-in-Aid for Research Activity Start-up
Spectral properties of interface problems for Maxwell systems
麦克斯韦系统界面问题的谱特性
- 批准号:
EP/W006553/1 - 财政年份:2022
- 资助金额:
$ 16.44万 - 项目类别:
Research Grant
Geometric analysis of partial differential equations and inverse problems
偏微分方程和反问题的几何分析
- 批准号:
22K03381 - 财政年份:2022
- 资助金额:
$ 16.44万 - 项目类别:
Grant-in-Aid for Scientific Research (C)