Unique Continuation for Geometric Wave Equations, and Applications to Relativity, Holography, and Controllability

科学、经济和工程中的各种现象都是由偏微分方程（PDE）数学建模的。波动方程是偏微分方程的一个重要子类，存在于许多基本的物理方程中，如麦克斯韦方程（电磁学）、杨-米尔斯方程（粒子物理学）、爱因斯坦场方程（引力）和欧拉方程（流体动力学）。特别是，爱因斯坦方程中隐藏的波动结构导致了在2016年实验检测到引力波之前的世纪对引力波的预测。偏微分方程的一个基本问题，特别是波动方程，是在给定适当数据的情况下找到唯一的解。这可以被解释为能够在给定系统的初始条件下“预测未来”。另一方面，在方程可能不总是解的情况下，仍然有必要问解是否保持唯一，如果它们存在的话;这就是唯一连续性的问题。直观地说，这个问题是问初始数据和解之间是否存在一一对应。现在有大量文献围绕唯一延拓理论。现代发展始于1939年Carleman的工作，并继续由Calderón，Hörmander，Tataru和许多其他人突破。主要的分析技术是一类加权不等式，现在被称为Carleman估计。我最近在这个方向的贡献围绕着独特的连续性的线性和非线性波;这也形成了我提出的研究的骨干。重点是研究“退化”的设置，其中经典理论不适用。另一个重要的目标是发展强大的几何技术，适用于各种各样的弯曲settings.The大部分拟议的研究计划涉及应用的结果和技术，这一独特的延续理论对其他问题。在过去的几十年里，唯一延拓与偏微分方程和物理学的其他方面之间存在着许多联系。例如，波动方程的唯一延拓结果最近已被应用于相对论中的对称性扩展和刚性结果。此外，Carleman估计，特别是对波动方程，一直是一个宝贵的工具，研究逆和控制理论问题的背景下，偏微分方程和微分几何。一个正在进行的研究项目，一个主要组成部分，拟议的计划，是应用独特的延续技术，研究全息原理的理论物理。这在很大程度上是由AdS/CFT对应激发的，它大致假定了反德西特时空中的引力动力学与其边界上的共形场论之间的对应。虽然这一思想在理论物理学中有着巨大的影响力，但在严格的数学公式和结果方面却进展甚微。这个项目的最终目标是为这些物理思想发展数学基础。在这方面的努力，在经典相对论的背景下，第一个主要步骤，是制定和证明一个对应的声明作为一个独特的延续问题的爱因斯坦场equations.Another主要方面的研究是应用这些技术，特别是新的Carleman估计，对其他问题的几何偏微分方程。一个例子是可控性的问题：这是问一个人是否可以使用有限的控制（例如，通过边界数据）将一个系统（例如，由偏微分方程建模）驱动到一个优选的状态。另一个感兴趣的领域是逆问题，它询问是否可以通过仅进行有限的测量（例如，解的边界值）来确定系统（数学上，PDE）。鉴于波在物理学中的普遍性，波的控制和逆问题与科学和工程中的重要问题密切相关。

A wide variety of phenomena in science, economics, and engineering are mathematically modelled by partial differential equations, or PDEs. Wave equations form an important subclass of PDEs and are found within many fundamental equations of physics, such as the Maxwell equations (electromagnetics), Yang-Mills equations (particle physics), Einstein field equations (gravitation), and Euler equations (fluid dynamics). In particular, the wave structure hidden within the Einstein equations led to the prediction of gravitational waves a century before their experimental detection in 2016.A basic problem in PDEs, and for wave equations in particular, is to find a unique solution given appropriate data. This can be interpreted as being able to "predict the future" given initial conditions for a system. On the other hand, in settings where the equation may not always be solved, it remains pertinent to ask whether solutions, if they exist, remain unique; this is the problem of unique continuation. Intuitively, this question asks whether there is a one-to-one correspondence between initial data and solutions.There is now significant literature surrounding the theory of unique continuation. Modern developments began with the work of Carleman in 1939 and continued with breakthroughs by Calderón, Hörmander, Tataru, and many others. The main analytic technique is a class of weighted inequalities now known as Carleman estimates.My recent contributions in this direction revolve around unique continuation properties for linear and nonlinear waves; this also forms the backbone of my proposed research. The focus is on studying "degenerate" settings, for which the classical theory fails to apply. Another essential goal is the development of robust geometric techniques that apply to a wide variety of curved settings.The bulk of the proposed research programme deals with applying the results and techniques of this unique continuation theory toward other problems. In the past decades, there have been numerous connections between unique continuation and other aspects of PDEs and physics. For instance, unique continuation results for wave equations have been recently applied in relativity toward symmetry extension and rigidity results. Furthermore, Carleman estimates, in particular for wave equations, have been an invaluable tool for studying inverse and control theory problems in the context of PDEs and differential geometry.One ongoing research project, a major component of the proposed programme, is to apply unique continuation techniques to study holographic principles in theoretical physics. This is largely motivated by the AdS/CFT correspondence, which roughly posits a correspondence between gravitational dynamics in Anti-de Sitter spacetime and conformal field theories on its boundary. While this idea has been tremendously influential in theoretical physics, there has been scant progress in terms of rigorous mathematical formulations and results. This project ultimately aims to develop mathematical underpinnings to these physical ideas. A first major step in this endeavour, in the context of classical relativity, is to formulate and prove a correspondence statement as a unique continuation problem for the Einstein field equations.Another major aspect of my research is to apply these techniques, novel Carleman estimates in particular, toward other problems for geometric PDEs. One example is the question of controllability: this asks whether one can drive a system (say, modelled by PDEs) to a preferred state using limited controls (for instance, through boundary data). Another area of interest is inverse problems, which asks whether one can determine a system (mathematically, a PDE) by making only limited measurements (for instance, the boundary values of solutions). Given the prevalence of waves in physics, both control and inverse problems for waves are well-connected to important questions in science and engineering.