Selected topics in harmonic analysis

在真实的生活中遇到的大多数结构都是由较简单的部件组成的复杂组件。有效的分析需要仔细地分解这样一个对象，这样整体的属性就可以转化为碎片，反之亦然。谐波分析使这种分解机制精确。它的影响跨越了数学内外的学科-它是量子物理学和许多现代技术（如信号处理，医学和地震成像）的首选语言。在这个建议中的问题，虽然植根于调和分析，在于它的接口领域，如几何测度理论，加法组合，偏微分方程（PDE）和几个复杂的变量，解决问题的兴趣，以多个数学社区。各级都有研究培训的机会。

Most structures encountered in real life are complex assemblies of simpler components. Effective analysis requires careful decomposition of such an object, so that properties of the whole can be translated to the pieces and vice versa. Harmonic analysis makes such decomposition mechanisms precise. Its impact spans disciplines within and beyond mathematics - it is the language of choice in quantum physics and many modern technologies like signal processing, medical and seismic imaging. The problems in this proposal, while rooted in harmonic analysis, lie at its interface with areas such as geometric measure theory, additive combinatorics, partial differential equations (PDE) and several complex variables, addressing issues of interest to multiple mathematical communities. Opportunities for research training exist at all levels.