Three problems in Harmonic Analysis

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In this mathematics research project, the focus is on three different problems in harmonic analysis. The first one is on a divergence-curl type estimate on a large class of non-abelian homogeneous Lie groups that includes the Heisenberg group as the simplest, nontrivial model. The second one is about the construction of a certain class of pseudodifferential operators that are adapted to two different flags; these operators will then be used to study the solution operators of certain partial differential equations that arise naturally in the study of several complex variables. The third problem is about a class of nonlinear wave equations whose elliptic parts exhibit conformal invariance, and we bring in elements from conformal geometry in studying these wave equations. These problems present interesting challenges from the analytical point of view.Broadly speaking, harmonic analysis studies functions or signals by decomposing them into different frequencies. Its applications are abundant, ranging from signal processing and tomography to problems arising from physics such as relativity and quantum mechanics. Yung and his collaborators will investigate some deep and fundamental questions in harmonic analysis, along with its connections to other areas of mathematics such as several complex variables and conformal geometry. In particular, in one of the proposed projects, a study of how certain nonlinear waves propagate through space will be carried out. The research will be in collaboration both with faculties that are experts in the field, as well as some young researchers who are just starting their research careers.