Problems in Time Frequency Analysis

这个数学研究项目旨在回答调和分析中的问题，这些问题在现代分析中非常感兴趣。首先，德米特提出调查的基本问题，是否相空间转换的各类功能是线性无关的。Demeter和他的合作者们最近提出了两个新的工具：一个是随机Schroedinger算子的谱理论，另一个是同时丢番图逼近理论。预计进一步的分析将揭示与其他数学领域的联系。其次，在过去的五年中，关于沿着向量场的奇异积分的可微性和有界性的问题获得了巨大的动力。在他最近的工作的基础上，德米特提出使用分析和组合的方法来研究各种Kakeya样的极大算子，并获得希尔伯特变换沿平面中给定数量的方向沿着的最佳界。德米特还将研究双线性傅立叶限制域的曲率。除其他事项外，这涉及扩展已知的双线性希尔伯特变换的一致估计。本研究计划将使用调和分析、组合数学、数论、谱理论和遍历理论等数学领域的各种工具，出版本研究计划的成果将有助于数学界了解调和分析、数论和遍历理论等数学研究领域之间的重要联系。海尔-拉马纳坦-托皮瓦拉猜想的解决可能会对各种科学产生重大影响，例如信号处理，其中伽柏和小波分析发挥了重要作用。该研究项目包括一个人力资源部分，包括培训研究生和博士后研究人员。

This mathematics research project aims at answering questions in harmonic analysis that are of great interest in modern analysis. First, Demeter proposes to investigate the fundamental problem of whether the phase space translates of various classes of functions are linearly independent. Together with his collaborators, Demeter has recently introduced two new tool: One is the spectral theory of random Schroedinger operators, the other one is the theory of simultaneous Diophantine approximation. It is expected that further analysis will reveal connections with various other fields of mathematics. Second, the question of differentiability and boundedness of singular integrals along vector fields has gained a huge momentum over the last five years. Building on his recent work, Demeter proposes to use analytic and combinatorial methods to investigate various Kakeya-like maximal operators, and to obtain optimal bounds for the Hilbert transform along a given number of directions in the plane. Demeter will also investigate the bilinear Fourier restrictions to domains with curvature. Among other things, this involves extending known uniform estimates for the bilinear Hilbert transform. This research project will employ a wide variety of tools from diverse areas of mathematics such as harmonic analysis, combinatorics, number theory, spectral theory and ergodic theory.The publication of the work that is expected to arise from this mathematics research project will enhance the mathematical community's understanding of the important connections among various mathematics research areas, such as harmonic analysis, number theory and ergodic theory. The resolution of the Heil-Ramanathan-Topiwala conjecture could have a significant impact on various sciences, such as signal processing, where Gabor and wavelet analysis play an important role. This research project includes a human resources component consisting of training graduate students and postdoctoral researchers.