Selected topics in harmonic analysis

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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. ******1. Patterns in sets:******Identifying patterns in large but otherwise arbitrary sets is currently a vibrant area of research both in pure mathematics and data science. We aim to build a cohesive theory that explains numerous phenomena concerning configurations in sets, and places them in the context of fractal geometry and Euclidean harmonic analysis on manifolds, while highlighting similarities and differences between the discrete and continuous counterparts. ******2. Sets of Directions:******Originating from deep open questions in the field such as the Euclidean Kakeya, Bochner-Riesz and restriction conjectures, this genre of problems focuses on behaviour of operators that rely on line segments in many directions. The proposal specifically addresses directional maximal operators, maximal directional Hilbert transforms and their relations with Kakeya-like sets. ******3. Smoothing and Decoupling:******Decoupling inequalities are currently at the cusp of research in multiple areas. They have proven instrumental in the resolution of long-standing problems in number theory and geometric analysis. Our objective is to explore applications of decoupling in incidence geometry and PDE. ******4. Resolution of singularities:******Zero sets of functions play a key role in many problems in geometry and analysis. The behaviour of many singular and oscillatory integrals of interest depends on the microlocal***structure of these sets. We use an algorithm for resolution of singularities developed earlier to study oscillatory integral operators and jumping numbers of real-analytic functions. ******5. Cauchy integral and Menger curvature:******The Cauchy integral on a graph is amazingly versatile: it is a reproducing formula for holomorphic functions, a fundamental example of a Calderon-Zygmund singular integral operator and embodies a geometric quantity called the Menger curvature. Do similar analytic and geometric characterizations hold for other kernels of interest? We propose a curvature-based approach for studying certain holomorphic reproducing kernels arising in multidimensional complex analysis.