Nonlinear hyperbolic differential equations and Fourier analysis

我计划解决以下具体问题。 首先，我想证明新的哈茨估计不使用显式参数。 这可能会大大扩大其应用范围。 我还想证明拟线性波动方程的几乎整体存在性，并研究非线性波动方程在非捕获障碍物外是否存在整体存在性。 我想使用这些项目中开发的技术来研究相对论中的问题，包括史瓦西解的稳定性。 最后，我想继续我的工作在调和分析的问题，出现在几何情况下。上述问题自然产生的数学和物理领域之间的相互作用，包括广义相对论，量子力学和量子混沌。 所采用的技术包括固定相位和奇点传播的研究。 在各大大学的量子物理学小组中，有一群非常活跃的研究人员在研究高能本征态，我特别有兴趣在这一领域做出进一步的贡献。

I plan on working on the following specific problems. First I would like to prove new Strichartz estimates without using explicit parametrices. This could potentially greatly widen the scope of their applications. I would also like to prove almost global existence for quasilinear wave equations and investigate whether there is global existence for nonlinear wave equations outside of nontrapping obstacles. I would like to use the techniques developed on these projects to study problems in relativity theory, including the stability of the Schwarzschild solution. Finally, I would like to continue my work on problems in harmonic analysis that arise in geometrical situations.The above problems arise naturally from interactions between mathematics and areas in physics that include general relativity, quantum mechanics, and quantum chaos. The techniques employed include stationary phase and the study of propagation of singularities. There is a very active group of researchers in quantum physics groups at major universities studying high-energy eigenstates, and I am especially interested in making further contributions to this area.