Geometric-Analytic Studies of the Einstein Equations and Other Partial Differential Equations

该项目涉及数学和物理领域的若干主题，以及这些领域之间的相互作用。这些主题的主要目的是推动我们在数学和物理方面的知识。特别是，这涉及几何分析和非线性偏微分方程(PDE)，重点是广义相对论中的爱因斯坦方程。这个项目的新见解和新方法对于解决其他结构相似的偏微分方程也将是重要的，其中一些是科学和技术模型的核心。PI研究的一个主要分支与爱因斯坦方程有关，爱因斯坦方程将我们宇宙的物理内容与几何联系起来，因此它们赋予了物理定律一个几何结构。这些方程是广义相对论的核心，广义相对论在很大程度上支配着我们宇宙的物理学。探索这些方程式将有助于更好地了解整个宇宙以及星系、双星黑洞或双星中子星等孤立系统。2015年，高级LIGO(ALIGO)项目对引力波的观测标志着一个新时代的开始，来自我们宇宙遥远区域的信息直接从宇宙本身(而不是从电磁波，如望远镜中的光)解码。要解开新的结构，比以往任何时候都更需要数学之间的协同作用，特别是对偏微分方程和几何学、物理学和天体物理学的分析。PI将在她的结果的基础上开发新的方法来实现这些目标。通过这个项目的教育部分，国际和平协会的研究还将通过教学和外联活动在更广泛的意义上产生直接影响。国际和平协会将培训这些领域的学生和博士后，并通过广泛的外联活动接触到公众，包括代表性不足的群体。PI还将通过出版物、会议和互联网交流她的结果。PI将开发和探索新的数学方法来研究爱因斯坦方程和其他描述物理现象的非线性偏微分方程组。特别是，PI建议研究：(1)专注于具有辐射的时空的爱因斯坦方程的柯西问题；(2)引力波的数学及其记忆效应(广义相对论中的时空永久变化显示为探测器中测试质量的永久位移)，以及其他物理理论中的记忆类似物；以及(3)欧拉方程和其他PDE本身及其与爱因斯坦方程的耦合。虽然所有这些主题都围绕着一种纯粹的数学(几何分析)处理，但每一个主题都会产生可以直接应用于物理实验的结果。建议的研究的一部分将继续PI之前的工作，将她的数学见解与实验联系起来(特别是aLIGO)。预计引力波记忆效应将在不久的将来被测量出来。这些新项目将部分建立在国际和平倡议和合作者的最新成果和方法的基础上，但也需要新的想法和方法。派和D·加芬克尔推导出了两个电磁记忆的类似物，从而第一次超出了广义相对论的范畴。PI和合作者将继续他们的研究，以完成对广义相对论中记忆的理解，并将他们的研究扩展到其他物理理论(例如，量子电动力学)。PI将把这条研究路线与她关于全局柯西问题的另一个项目联系起来，她将用几何分析方法研究其他偏微分方程组。此外，爱因斯坦方程与其他偏微分方程组的解将在一般情况下进行研究，包括渐近平坦的时空和宇宙时空，因此有望发现有趣的局部和全球结构。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project concerns a number of topics in mathematics and physics as well as at the interplay between these fields. The overarching aim of these topics is to push forward our knowledge in mathematics as well as in physics. In particular, this involves geometric analysis and nonlinear partial differential equations (PDE) with a focus on the Einstein equations in general relativity. The new insights and methods from this project will also be important to solve other structurally similar PDE, some of which are central to models in science and technology. One main branch of the PI's research concerns the Einstein equations linking the physical content of our universe to geometry, thus they give the physical laws a geometric structure. These equations are at the heart of the theory of general relativity, which governs the physics of our universe in the large. Exploring these equations will lead to a better understanding of the universe as a whole and of isolated systems such as galaxies, binary black holes or binary neutron stars. The observation of gravitational waves by the advanced LIGO (aLIGO) project in 2015 marked the beginning of a new era where information from distant regions of our universe is decoded directly from the universe itself (rather than from electromagnetic waves like for instance light in telescopes). More than ever, synergies between mathematics, in particular analysis of PDE and geometry, physics and astrophysics are needed to unravel the new structures. The PI will build on her results to develop new methods to achieve these goals. Through the educational component of this project, the PI's research will also have direct impact in a wider sense via teaching and outreach activities. The PI will train students and postdocs in these fields, and through broad outreach activities reach members of the public including underrepresented groups. The PI will also communicate her results through publications, conferences and the internet.The PI will develop and explore new mathematical methods to investigate the Einstein equations and other nonlinear PDE describing physical phenomena. In particular, the PI proposes to study: (1) the Cauchy problem for the Einstein equations focussing on spacetimes with radiation; (2) the mathematics of gravitational waves and their memory effect (a permanent change of the spacetime showing as a permanent displacement of test masses in a detector like aLIGO) in general relativity, as well as analogs of memory in other physical theories; and (3) Euler equations and other PDE per se and their coupling to the Einstein equations. Whereas all these topics center around a purely mathematical (geometric-analytic) treatment, each one of them will produce results that can be directly applied in physics experiments. Parts of the suggested research will continue the PI's former work linking her mathematical insights to experiments (aLIGO in particular). It is expected that the gravitational wave memory effect will be measured in the near future. These new projects will partially build on the PI's and collaborators' recent results and methods but will also require new ideas and approaches. The PI and D. Garfinkle derived two analogs of memory in electromagnetism, thus for the first time outside of general relativity. The PI and collaborators will continue their research to complete the understanding of memory in general relativity, and to extend their research to other physical theories (e.g., quantum electrodynamics). The PI will connect this line of research with her other project about the global Cauchy problem, and she will study other PDE with geometric-analytic methods. Moreover, solutions of the Einstein equations coupled to other PDE will be investigated for general situations, including asymptotically flat as well as cosmological spacetimes, and thereby interesting local and global structures are expected to be found. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.