Selected topics in harmonic analysis

现实生活中遇到的大多数结构都是由较简单组件组成的复杂组件。有效的分析需要仔细分解这样的对象，以便可以将整体的属性转换为碎片，反之亦然。调和分析使这种分解机制变得精确。它的影响跨越了数学内外的学科--它是量子物理以及信号处理、医学和地震成像等许多现代技术的首选语言。这项建议的问题，虽然根植于调和分析，但存在于其与几何测度论、加性组合学、偏微分方程(PDE)和几个复变量等领域的接口，解决了多个数学社区感兴趣的问题。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.