Spectral Problems in Geometry and Partial Differential Equations

本项目将集中于偏微分方程谱理论的三个领域：（1）几何散射中的共振，（2）两步幂零李群的调和分析，以及（3）非线性色散方程理论中的逆谱方法。在第一个项目中，主要研究人员将继续他的工作，对光谱和散射理论的渐近双曲流形和复杂的流形，研究散射共振的性质及其与基本几何不变量的关系。在第二个项目中，他将致力于理解两步幂零李群的热核和波迹奇点的渐近行为。具有奇异初值的非线性色散方程，通过逆散射方法，可以看作是一个非常奇异的势的散射数据的线性化流。我们希望将KdV和mKdV方程的逆散射图像扩展到这样的奇异数据，并通过在Hilbert空间上构建流来更深入地了解这些动力学系统，这些初始数据是奇异的，但从动力学的角度来看是非常自然的。偏微分方程为波的传播和热流等各种物理现象提供了基本的数学模型。 任何物理模型的一个同样重要的部分是它所描述的系统的几何形状：例如，不同形状的乐器产生不同频率的声波，即使在所有情况下波的产生和传播都由相同的微分方程控制。上述前两个项目研究了某个偏微分方程的解与定义解的域的几何之间的相互作用，并试图将解的可量化性质与底层几何的可量化性质联系起来。这些项目是数学界更大努力的一部分，以理解“逆问题”，其中物理系统的几何形状是从可测量的数据重建的，这些数据是偏微分方程的解：地震学和医学成像是应用数学领域中出现这种逆问题的地方。第三个项目是对物理学中产生的一类方程的研究的贡献，这些方程通过将强大的求解方法扩展到更丰富的数据集来描述非线性波的传播。

This project will focus on three areas of spectral theory for partial differential equations: (1) resonances in geometric scattering, (2) harmonic analysis on two-step nilpotent Lie groups, and (3) inverse spectral methods in the theory of nonlinear dispersive equations. In the first project, the principle investigator will continue his work on spectral and scattering theory for asymptotically hyperbolic manifolds and complex manifolds, studying properties of scattering resonances and their relationship to underlying geometric invariants. In the second project, he will work toward understanding the asymptotic behavior of heat kernels and singularities of the wave trace for two-step nilpotent Lie groups. Nonlinear dispersive equations with singular initial data may be viewed, via the inverse scattering method, as linearized flows for the scattering data of a very singular potential. We hope to extend the inverse scattering picture for the KdV and mKdV equations to such singular data and obtain greater insight into these dynamical systems--by constructing the flow on Hilbert spaces of initial data which are singular but very natural from a dynamical point of view.Partial differential equations provide underlying mathematical models for such diverse physical phenomena as wave propagation and heat flow. An equally important part of any physical model is the geometry of the system it describes: for example, differently-shaped musical instruments produce sound waves with different frequencies, even though in all cases the production and propagation of waves is governed by the same differential equation. The first two projects above study the interaction between solutions of a certain partial differential equation and the geometry of the domain where the solutions are defined, and seek to relate quantifiable properties of the solutions to quantifiable properties of the underlying geometry. These projects are part of a larger effort in the mathematical community to understand "inverse problems" in which the geometry of a physical system is reconstructed from measurable data which are the solutions of a partial differential equation: seismology and medical imaging are among the areas of applied mathematics where such inverse problems occur. The third project is a contribution to the study of a class of equations arising in physics which describe the propagation of nonlinear waves by extending powerful solution methods to a richer set of data.