Methods of Harmonic Analysis in Partial Differential Equations and Integrable Systems

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The proposed research suggests investigations in the area of Harmonic Analysis and applications of its methods to Partial Differential Equations and to the Theory of Integrable Systems. The proposer will apply Littlewood-Paley methods to develop a unified theory of matrix-weighted function spaces and will also develop the method of level sets to characterize the estimates on Fourier Integral Operators with singularities. She will apply the above techniques to study the solutions of semilinear hyperbolic PDEs in the situation when caustics appear and to study analytic properties of the Riemann-Hilbert problem that arise in the Scattering Theory, in the context of completely integrable systems.Harmonic analysis investigates complex behavior of a system by representing it as a sum of basic elements (wavelets, atoms, Fourier modes, etc.) behaving in a simple way. This decomposition becomes the central object of studies in Modern Analysis. The tools of Harmonic Analysis are widely used in applications which describe natural phenomena in optics, scattering theory, quantum physics, andcombinatorics. Results in this direction will have immediate applications in other areas of mathematics and other natural sciences, primarily engineering, physics, and biology. Central achievements of Harmonic Analysis are widely embraced in medicine and other disciplines directly benefitting well-being of humankind.