Carleson's Theorem in Analysis, Scattering, and Ergodic Theory

波包分析在数学分析的各个领域都有应用：散射理论、经典傅立叶分析、遍历理论。该项目将扩大波包分析在这些领域的应用，从而加深对这些领域的理解。在这个项目中考虑的特殊问题是关于傅里叶级数收敛的Carleson定理的自然非线性变体，在波包分析及其各种应用背景下的二次傅里叶分析的发展，以及超越布尔甘返回时间定理的遍历平均的各种收敛定理。由于它在数学分析领域的许多应用，该项目有助于促进我们在各个方向上对该领域的理解，因此对几个研究领域具有潜在的影响。在过去，与这个项目密切相关的分析已经在非数学世界中找到了自己的应用方式，最著名的是通过90年代计算数学中的“小波”革命。重要的是保持我们在该领域的基础研究，为未来的应用奠定基础。此外，加州大学洛杉矶分校在分析领域保持着一个活跃的小组，培养了许多该领域的年轻数学家，并成功地将他们安置在学术界。该项目有助于继续为未来的数学劳动力做出贡献。

Wave packet analysis has applications to various fields within mathematical analysis: scattering theory, classical Fourier analysis, ergodic Theory. This project will broaden the use of wave packet analysis in these areas and thus lead to deeper understanding of these areas.Particular problems considered in this project are a natural nonlinear variant of Carleson's theorem on convergence of Fourier series, the development of quadratic Fourier analysis in the context of wave packet analysis and its various applications, and various convergence theorems for ergodic averages going beyond Bourgain's return times theorem.With its many applications to fields within mathematical analysis, the project helps to advance our understanding of the field in a variety of directions and thus has potential impact on several lines of research. In the past, analysis closely related to this project has found its way into applications to the non-mathematics world, most notably through the "wavelet" revolution in computational mathematics in the nineties. It is important to maintain our basic research in the field to lay out the groundwork for future applications. In addition, UCLA maintains an active group in analysis, training many young mathematicians in the area and successfully placing them in academia. This project helps continuing this contribution to the future mathematical workforce.