Linear Partial Differential Equations on Singular Spaces

在这个奖项PI将研究微分算子流形奇异度量结构。一个项目是研究奇异空间上的波动方程。梅尔罗斯、瓦希和PI将研究复杂奇异几何中波动方程解的衍射，最终可能是一大类分层空间。这建立在以前的工作与梅尔罗斯，其中表明，一个奇点的解决方案的波动方程相互作用的锥点，以产生一个“衍射”球面波阵面从锥点发出。如果入射波前没有太精确地聚焦在锥点上，衍射波前的奇异性可能比入射波前的奇异性弱。另一个研究方向涉及流形上的薛定谔方程，其中无限的传播速度有一些有趣的结果。PI将研究在捕获几何中薛定谔演化的色散平滑效应方面，首先关注与拉格朗日子流形相关的规则性的传播。量子力学数学理论中的一个中心问题是：粒子的经典动力学与其相应的量子态之间的关系是什么？对这个问题的理解不仅来自于对量子力学能量算符或“哈密顿量”本身的直接研究，而且来自于涉及它的其他基本方程，如热方程、波动方程，当然还有含时薛定谔方程。因此，本研究的重点包括几类偏微分方程的几何分析。一个项目研究当波与尖角（某种概括）相互作用时波会发生什么--由于衍射的影响，波前如何移动的几何形状在这些情况下可能非常微妙。 可能的物理应用包括“逆问题”，其中一个试图推断出一个物体的结构（例如地球的内部）通过观察波已经通过它。另一个项目是研究薛定谔方程的解在弯曲空间的行为;这样的解决方案描述了量子粒子的时间演化。 这一课题的进展也可能对非线性薛定谔方程产生影响，该方程出现在非线性光学和玻色-爱因斯坦凝聚理论以及其他物理应用中。

Under this award the PI will study differential operators on manifolds with singular metric structures. One project is to study the wave equation on singular spaces. Melrose, Vasy, and the PI will investigate the diffraction of solutions of the wave equation by complex singular geometries, perhaps eventually a large class of stratified spaces. This builds on previous work with Melrose, which showed that a singularity of a solution of the wave equation interacts with a cone point to produce a "diffracted" spherical wavefront emanating from the cone point. The singularity of the diffracted front may be weaker than that of the incident front if the latter is not too precisely focused on the cone point. Another direction of research involves the Schroedinger equation on manifolds, where the infinite speed of propagation has some intriguing consequences. The PI will investigate aspects of the dispersive smoothing effect for Schroedinger evolution in trapping geometries, focusing initially on the propagation of regularity associated to Lagrangian submanifolds.A central question in the mathematical theory of quantum mechanics is: what is the relationship between the classical dynamics of a particle and its corresponding quantum states? Insights into this problem have come not only from the direct study of the quantum mechanical energy operator or "Hamiltonian" itself, but also from other fundamental equations involving it, such as the heat equation, the wave equation, and, naturally, the time-dependent Schroedinger equation. The focus of this research consequently includes the geometric analysis of several kinds of partial differential equations. One project investigates what happens to waves when they interact with (a certain generalization of) sharp corners---the geometry of how wavefronts move can be quite subtle in these cases owing to the effects of diffraction. Possible physical applications include "inverse problems" in which one attempts to deduce the structure of an object (e.g. the interior of the earth) from observation of waves that have passed through it. Another project is to study the behavior of solutions of the Schroedinger equation on curved spaces; such solutions describe the time-evolution of a quantum particle. Progress in this subject may also have consequences for the nonlinear Schroedinger equation, which arises in nonlinear optics and the theory of Bose-Einstein condensates, among other physical applications.