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

拟议的研究将涉及不同数学领域之间的问题：调和分析、遍历理论和数论。这些问题发生在不同的背景下，然而，某些指数和的性质以及分析数论的相关方法发挥了核心作用，例如指数和的Hardy-Littlewood方法。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。这些都是欧氏空间上调和分析问题的离散类比，也自然地与非对易变换的逐点遍历理论相联系。从技术上讲，“圈方法”的一个关键作用是将“圈方法”扩展到一个算子值设置，这是由于基础群的非对易性而产生的。另一组提出的问题是在加法数论/组合学的背景下，因为它们表明在正密度的子集中或在整数格的有限划分后得到的单元中存在某些结构。重点是傅立叶分析方法，以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几种技术，其中一些是最新的，仍在开发中。重点是上述领域技术的相互作用，以解决感兴趣的公开问题。该项目将使PI能够继续支持研究生和博士后，并总体上将年轻的研究人员介绍给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.