Variable coefficient Fourier Analysis and its applications

摘要本文讨论了黎曼流形（包括紧流形和非紧流形）上的波动方程的估计问题。 我们感兴趣的几何形状，边界和规则的度量影响某些基本估计。 这类问题出现在流形上的调和分析研究、非线性波动方程的局部和全局解的研究以及量子混沌中本征函数的研究中。 虽然这些主题在其物理和历史起源上是广泛分离的，但相关的数学是密切相关的。 各个领域的技术和见解以富有成效的方式相互交叉。 特别是，当前许多研究的一个共同主题（以及本提案中的问题）是试图理解和利用线性和非线性波动方程的本征函数和解的质量集中。我们所考虑的基本估计是本征函数和准模在空间中的勒贝格空间估计（线性和双线性），以及时空中的（局部或全局）哈茨估计。 主要问题集中在几何形状，特别是边界的存在如何影响估计和饱和它们的解决方案的种类。 后一个问题是密切相关的研究（但仍然没有很好地理解）的浓度，振荡和大小的频谱渐近的模式和准模式的性质的问题。 在非紧的情况下，它也与共振态的分布以及共振态与被困测地线的关系密切相关。上述问题自然地产生于数学与物理学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。

ABSTRACTThis proposal is concerned with estimates of wave equations on (both compact and non-compact) Riemannian manifolds, possibly with boundary. We are interested in how the geometry, the boundary and the regularity of the metric influence certain basic estimates. Problems of this kind arise in the study of harmonic analysis on manifolds, the study of local and global solutions of nonlinear wave equations and in the study of eigenfunctions in quantum chaos. Although these topics are widely separated in their physical and historical origins, the relevant mathematics is closely related. Techniques and insights in the various areas cross-fertilize each other in a fruitful way. In particular, a common theme of much current research (and the problems in this proposal) is to try to understand and exploit the mass concentration of eigenfunctions and solutions of linear and nonlinear wave equations. The basic estimates that we have in mind are Lebesgues-space estimates (both linear and bilinear) in space for eigenfunctions and quasi-modes, and (local or global) Strichartz estimates in space-time. The main questions center around how the geometry and especially the presence of a boundary affects the estimates and the kinds of solutions that saturate them. The latter issue is closely related to the much studied (but still not well understood) questions of concentration, oscillation and size properties of modes and quasi-modes in spectral asymptotics. In the non-compact setting it is also closely related to the distribution of resonances and their relations to trapped geodesics.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.