Discrete problems in harmonic analysis with applications to ergodic theory and additive number theory

拟议的研究将处理的问题是在不同领域的数学之间的边界;调和分析，遍历理论和数论。这些问题发生在不同的背景下，但发挥核心作用的性质某些指数和相关的方法分析数论，如哈迪-利特尔伍德方法的指数和。一组问题涉及与离散幂零群上的多项式曲面相关的极大和奇异积分算子。这些都是离散类似的问题调和分析的欧几里德空间，也自然连接到逐点遍历理论的非交换变换。从技术上讲，一个至关重要的作用是由“圆方法”的扩展到一个运营商值的设置，这是由于非交换性的基础组。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某些结构的存在，或者在整数格有限划分后获得的单元中。重点是傅立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。它旨在利用和扩展几种技术，其中一些是最近的，仍在开发中。重点是上述领域的技术的相互作用，以攻击感兴趣的开放问题。该项目将使PI能够继续支持研究生和博士后，并在一般情况下，介绍年轻的研究人员到这个快速发展的数学领域。

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.