Linear Partial Differential Equations on Singular Spaces

在这个奖项下，PI将研究具有奇异度量结构的流形上的微分算子。一个项目是研究奇异空间上的波动方程。Melrose， Vasy和PI将通过复杂的奇异几何来研究波动方程解的衍射，也许最终会研究一大类分层空间。这是建立在Melrose之前的工作基础上的，该工作表明，波动方程解的奇点与锥点相互作用，产生从锥点发出的“衍射”球形波前。如果入射锋不太精确地聚焦在锥点上，那么衍射锋的奇异性可能比入射锋的弱。另一个研究方向涉及流形上的薛定谔方程，其中无限传播速度有一些有趣的结果。PI将研究色散平滑效应在捕获几何薛定谔演化中的各个方面，首先关注与拉格朗日子流形相关的正则性传播。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？对这个问题的见解不仅来自对量子力学能量算符或“哈密顿算符”本身的直接研究，还来自与之相关的其他基本方程，如热方程、波动方程，当然还有时变薛定谔方程。因此，本研究的重点包括几种偏微分方程的几何分析。一个项目研究了当波与（某种泛化的）尖角相互作用时会发生什么——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相当微妙。可能的物理应用包括“逆问题”，其中人们试图通过观察穿过物体的波来推断物体的结构（例如地球的内部）。另一个课题是研究弯曲空间上薛定谔方程解的行为；这样的解描述了量子粒子的时间演化。这一学科的进展也可能对非线性薛定谔方程产生影响，它出现在非线性光学和玻色-爱因斯坦凝聚体理论中，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.