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
• 负责人: Akos Magyar
• 金额: $ 11.74万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0803190&HistoricalAwards=false
• 关键词: Discrete; problems; harmonic; analysis; applications

The proposed research will deal with problems that are on the boundary between different fields of mathematics; harmonic analysis, ergodic theory, and number theory. The problems occur in different contexts however a central role is played by properties certain exponential sums and related methods of analytic number theory, such as the Hardy-Littlewood method of exponential sums. One set of problems concern maximal and singular integral operators associated with polynomial surfaces on discrete nilpotent groups. These are discrete analogues of problems harmonic analysis on Euclidean spaces, and also are naturally connected to pointwise ergodic theory of non-commuting transformations. Technically, a crucial role is played by an extension of the "circle method" to an operator valued settings, which arises because of non-commutativity of the underlying group. Another set of the proposed problems are in the settings of additive number theory/combinatorics, as they are to show the existence of certain structures in subsets of positive density, or in cells obtained after a finite partitioning of integer lattices. The emphasis is on the Fourier analytic approach to address certain problems in the multidimensional or nonlinear settings.Because of its interdisciplinary nature, the project is expected to have an impact on the various fields involved. It aims utilize and extend several techniques, some of which is recent and is still being developed. The emphasis is on the interplay of techniques of the above fields, to attack open problems of interest. The project would enable the PI to continue to support graduate students and post-docs, and in general, to introduce young researchers to this rapidly developing area of mathematics.

拟议的研究将解决不同数学领域之间的边界问题；调和分析、遍历理论和数论。这些问题出现在不同的环境中，但核心作用是由某些指数和的属性和解析数论的相关方法（例如指数和的 Hardy-Littlewood 方法）发挥的。一组问题涉及与离散幂零群上的多项式曲面相关的最大和奇异积分算子。这些是欧几里得空间上调和分析问题的离散类似物，并且也自然地与非交换变换的逐点遍历理论相关。从技术上讲，将“循环方法”扩展到运算符值设置发挥着至关重要的作用，这是由于底层群的不可交换性而产生的。另一组提出的问题是在加法数论/组合学的设置中，因为它们要显示正密度子集中或整数格有限划分后获得的单元中某些结构的存在。重点是用傅里叶分析方法来解决多维或非线性环境中的某些问题。由于其跨学科性质，该项目预计将对所涉及的各个领域产生影响。它的目标是利用和扩展多种技术，其中一些是最近出现的并且仍在开发中。重点是上述领域技术的相互作用，以解决感兴趣的开放问题。该项目将使 PI 能够继续支持研究生和博士后，并总体上向年轻研究人员介绍这个快速发展的数学领域。