Some topics in time-frequency analysis

Lie和他的合作者将研究一些问题，这些问题位于时频分析和其他数学领域的接口，如加法组合学，几何测度理论，关联几何和偏微分方程。 在这个项目中，时频分析主题问题的主要目标是（a），发展一个令人满意的广义波包理论和（B），发展适当的环境，在更高的维度时频分析。在第一个方向上的一个步骤是由PI在他的博士学位。论文，在那里他介绍了一个新的角度对广义波包理论，后来证明有利于证明有界的多项式Carleson运营商在一个维度。 其中一些技术似乎是适用于研究的最大薛定谔算子在一维的有界性。 PI和他的合作者将调查其他几个问题，包括：有界的双线性希尔伯特变换沿着曲线;有界的高维版本的多项式Carleson运营商;逐点收敛的解决方案，以自由薛定谔方程在高维;收敛序列的部分傅立叶和。这个数学研究项目调查谐波分析中的各种问题，谐波分析是数学领域，对其他科学中出现的问题具有重要应用，例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。所采用的技术将连接几个数学领域，包括偏微分方程，关联几何和加法组合学。由此产生的协同作用将丰富分析工具和可用技术的武库;这些将在研究生研讨会上提出，旨在吸引学生和年轻研究人员到数学及其应用的这一动态领域。

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.