Variable Coefficient Fourier Analysis

这项研究涉及(紧致的和非紧致的)黎曼流形上波动方程的估计，可能是带边界的。研究人员感兴趣的是，度量的几何、边界和正则性如何影响某些基本估计。这类问题出现在研究流形上的调和分析，研究非线性波动方程的局部解和整体解，以及研究量子混沌中的本征函数。虽然这些主题在物理和历史起源上相去甚远，但相关的数学是密切相关的。各个领域的技术和洞察力相互促进，成果丰硕。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)Strichartz估计。主要问题围绕着几何，特别是边界的存在如何影响估计以及使估计饱和的解的种类。后一个问题与谱渐近中的模和准模的集中、振荡和尺寸性质等问题密切相关(但还不是很清楚)。在非紧凑背景下，它还与共振的分布及其与圈闭测地线的关系密切相关。在紧集下，我们还有兴趣探索本征函数的集中性质如何与其节点集相关，节点集是函数消失的点集。研究人员希望利用当前和新的本征函数估计来进一步推进Yau关于这些集合大小的猜想。他还有兴趣了解某些曲率假设如何影响这些估计，并探索与量子遍历的联系。上述问题自然产生于数学与物理领域(包括广义相对论、量子力学和量子混沌)之间的相互作用。所采用的技术包括静止相和奇点传播的研究。在主要大学的量子物理小组中，有一群非常活跃的研究人员研究高能本征态，研究人员特别有兴趣在这一领域做出进一步的贡献。

This research is concerned with estimates of wave equations on (both compact and noncompact) Riemannian manifolds, possibly with boundary. The investigator is 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 Lebesgue-space estimates (both linear and bilinear) in space for eigenfunctions and quasimodes, and (local or global) Strichartz estimates in spacetime. 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 quasimodes in spectral asymptotics. In the non-compact setting it is also closely related to the distribution of resonances and their relations to trapped geodesics. In the compact setting we are also interested in exploring how concentration properties of eigenfunctions are related to their nodal sets, which is the set of points where the function vanishes. The investigator wishes to use current and new estimates for eigenfunctions to make further progress on Yau's conjecture about the size of these sets. He also is interested in seeing how certain curvature assumptions affect these estimates and exploring connections with quantum ergodicity.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 the investigator is especially interested in making further contributions to this area.