Time-frequency analysis in small dimensions

这个项目研究时频分析，因为它第一次出现在二十世纪六十年代与Carleson的开创性工作几乎处处收敛的傅立叶级数。大约15年前，Carleson的技术被用来建立双线性希尔伯特变换的估计，这引发了时间-频率分析的快速发展。该项目的研究详细阐述了这一理论的重要方面，在小维度大于一个与其他分析领域相关的维度。二维双线性希尔伯特变换涉及到遍历理论中一个长期存在的公开问题，即两个交换变换的遍历平均几乎处处收敛。二维的时频分析也与Stein和Zygmund关于二维的希尔伯特变换和沿着向量场的微分的孪生理论有关。最后，低维李群上的傅立叶分析的非线性变体，它涉及到许多数学分支，如散射理论，引起了有趣的基本问题。它们中的一些是时频分析在小维度上的非线性变体。该项目的研究与数学物理有几个联系，可能会导致应用。研究了粒子沿沿着势散射的数学基础。它还研究了分析的基础沿着矢量场，其中提出了流体动力学的数学框架。这个项目在两个维度上的一些分析可以被解释为两个随机过程的特定相互作用，例如可以在金融数学中找到。这个项目将有助于继续在加州大学洛杉矶分校的谐波分析，这吸引了一些优秀的年轻数学家，并有助于保持劳动力在数学积极的研究和培训小组。该项目的另一个教育组成部分是继续为来自不同大学的研究生和博士后研究员举办一系列非常成功的暑期学校。

This project studies time-frequency analysis as it first appeared in the nineteen sixties with Carleson's seminal work on almost everywhere convergence of Fourier series. About fifteen years ago, Carleson's techniques were used to establish estimates for the bilinear Hilbert transform which initiated a rapid development of time-frequency analysis. The research of this project elaborates on important aspects of this theory in small dimensions greater than one with relevance to other areas of analysis. The bilinear Hilbert transform in two dimensions relates to a long standing open problem in ergodic theory about almost everywhere convergence of ergodic averages for two commuting transformations. Time-frequency analysis in two dimensions is also relevant to twin conjectures by Stein and Zygmund on Hilbert transform and differentiation along vector fields in two dimensions. Finally, a non-linear variant of Fourier analysis on low dimensional Lie groups, which relates to many branches of mathematics such as scattering theory, gives rise to interesting fundmental questions. Some of them are nonlinear variants of time frequency analysis in small dimensions. The research of this project has several connections to mathematical physics, which may lead to applications. It studies the mathematical foundations of scattering of particles along long range potentials. It also studies the foundations of analysis along vector fields, which present the mathematical framework of fluid dynamics. Some of the analysis of this project in two dimensions may be interpreted as a particular interplay of two random processes, such as can be found in financial mathematics. This project will help continue an active research and training group in harmonic analysis at UCLA, that has attracted a number of excellent young mathematicians and has helped maintain the workforce in mathematics. A further educational component of this project is to continue a series of very successful summer schools for graduate students and postdoctoral fellows in analysis from different universities.