Some topics in time-frequency analysis

李和他的合作者将研究时频分析和其他数学领域的一些问题，如加法组合学、几何测度论、关联几何和偏微分方程。本项目中以时频分析为主题的问题的主要目标是(A)发展一个令人满意的广义波包理论和(B)为更高维度的时频分析开发合适的环境。PI在他的博士论文中朝第一个方向迈出了一步，在那里他引入了一种关于广义波包理论的新观点，该理论后来被证明对证明多项式Carleson算子在一维有界性是有益的。其中一些技巧似乎适用于研究一维极大薛定谔算子的有界性。PI和他的合作者将研究其他几个问题，包括：双线性希尔伯特变换沿曲线的有界性；多项式Carleson算子的高维版本的有界性；高维自由薛定谔方程解的逐点收敛；部分傅立叶和序列的子序列的收敛。这个数学研究项目研究了调和分析中的各种问题，这是一个对其他科学中出现的问题有重要应用的数学领域，如计算机图形学中的模式识别；物理学中的电荷分布；地震学中的波传播。将使用的技术将跨越数学的几个领域，包括偏微分方程式、入射几何和加法组合学。由此产生的协同作用将丰富分析工具和现有技术的武器库；这些将在专门为吸引学生和年轻研究人员进入这一充满活力的数学及其应用领域而设计的研究生研讨会上介绍。

Lie and his collaborators will investigate a number of problems that lie at the interface of time-frequency analysis and other areas of mathematics such as additive combinatorics, geometric measure theory, incidence geometry and partial differential equations. The main objectives for the time-frequency analysis-themed problems in this project are (a), to develop a satisfactory generalized wave packet theory and (b), to develop the proper environment for the time - frequency analysis in higher dimensions. One step in the first direction was taken by the PI in his Ph.D. thesis, where he introduced a new perspective on the generalized wave packet theory which later proved beneficial in proving the boundedness of the Polynomial Carleson operator in dimension one. Some of these techniques appear to be applicable to the study of the boundedness of the maximal Schrodinger operator in dimension one. The PI and his collaborators will investigate several other problems, including: boundedness of the Bilinear Hilbert transform along curves; boundedness of the higher dimensional version of the Polynomial Carleson operator; pointwise convergence of the solution to the free Schrodinger equation in higher dimensions; convergence of the subsequences of sequences of partial Fourier sums. This mathematics research project investigates various problems in harmonic analysis, which is an area of mathematics that has important applications to problems that arise in other sciences, such as pattern recognition in computer graphics; electrical charges distribution in physics; wave propagation in seismology. The techniques to be employed will bridge several areas of mathematics, including partial differential equations, incidence geometry, and additive combinatorics. The resulting synergies will enrich the arsenal of analytical tools and available techniques; these will be presented in graduate seminars designed with the specific purpose to attract students and young researchers to this dynamic area of mathematics and its applications.