Methods of Harmonic Analysis in Partial Differential Equations and Integrable Systems

建议在调和分析领域进行研究，并将其方法应用于偏微分方程和可积系统理论。建议者将应用Littlewood-Paley方法来发展矩阵加权函数空间的统一理论，并将发展水平集方法来表征具有奇点的傅里叶积分算子的估计。她将运用上述技术研究焦散出现时半线性双曲偏微分方程的解，以及在完全可积系统的背景下研究散射理论中出现的黎曼-希尔伯特问题的解析性质。谐波分析通过将系统表示为以简单方式行为的基本元素（小波、原子、傅立叶模式等）的总和来研究系统的复杂行为。这种分解成为《现代分析》研究的中心对象。谐波分析工具被广泛应用于描述光学、散射理论、量子物理和组合学中的自然现象。这个方向的研究结果将直接应用于数学和其他自然科学领域，主要是工程学、物理学和生物学。谐波分析的核心成果被广泛应用于医学和其他学科，直接造福于人类的福祉。

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.