Geometric Analysis: Investigating the Einstein Equations and Other Partial Differential Equations

该项目涉及数学，物理及其接口的几个主题，重点是几何分析，偏微分方程（PDE）和广义相对论（GR）的研究。它旨在提高我们在数学和物理方面的知识。该项目的新见解和方法对解决其他结构相似的PDE也很重要。这些偏微分方程是科学和技术模型的核心，从物理学到生物学和化学，再到金融学、经济学或医学和心理学。独立于它们在各种应用中的根源，这些偏微分方程及其解经常表现出类似的有趣的结构，这些结构是通过几何分析的数学方法来研究的。因此，一个方向的数学结果可能会为解决完全不同领域或应用中的问题打开大门。该项目的一个主要研究方向是广义相对论中的爱因斯坦方程，这是宇宙的定律，将其物理内容与几何学联系起来。这一理论对GPS的工作也至关重要。研究这些方程将增加我们对宇宙的理解，无论是在大的还是在较小的区域，如星系，双中子星或双黑洞。除此之外，该项目不仅将回答物理学中的开放性问题，还将为物理模型创造新的想法。当两个大质量的物体，如中子星或黑洞合并时，就会产生引力波，并以时空涟漪的形式从源头穿过宇宙。2015年，先进LIGO（aLIGO）首次观测到了这种波，标志着一个新时代的开始，来自宇宙遥远区域的信息直接从宇宙本身解码（不同于望远镜）。解开新结构将依赖于数学、天体物理学和物理学之间的协同作用。该项目将建立在PI的先前成果的基础上，以开发新的方法来实现这些目标。该项目还将通过教学和外联活动产生更广泛的直接影响。PI将在这些领域培训学生和博士后，并通过广泛的推广活动，包括代表性不足的群体。PI将参加会议以传达结果。PI还将通过互联网和出版物提供结果。 在这个项目中，PI将开发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。PI将调查：（1）Einstein方程的Cauchy问题，主要集中在（I）具有辐射的时空，（II）当Einstein方程与物质系统耦合时，在GR中黑洞的形成，（2）引力波的数学，它们在GR中的记忆和相关效应以及其他物理理论中的类似记忆，（3）Euler方程和其他PDE。这些主要方向包括几个项目。他们中的许多人将依赖于PI以前的结果，但也需要新的想法和新的数学方法。PI最近对真空中的爱因斯坦方程和中微子辐射的结果揭示了引力辐射和记忆中的新结构的全景，预计将在当前和未来的引力波探测器中看到。计划中的部分研究将数学见解与实验（特别是LIGO/VIRGO）联系起来。此外，引力波记忆有望在不久的将来被探测到。PI和D。加芬克尔在电磁理论中推导出了两种类似的记忆。预计这些也将在实验中进行测量。数学方法广泛适用于从数学到物理的各种问题。PI和合作者将继续他们的研究，以完成对GR中引力辐射和记忆的理解，并将他们的研究扩展到其他物理理论。PI对GR中爱因斯坦方程的数学研究提供了解决其他PDE的几何分析方法。此外，新发现的结构在几何学和偏微分方程理论中也很重要。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project addresses several topics in mathematics, physics and their interface, focusing on geometric analysis, the study of partial differential equations (PDE) and general relativity (GR). It aims at enhancing our knowledge in mathematics as well as in physics. The new insights and methods from this project will also be important to solve other structurally similar PDE. These PDE are at the heart of models in science and technology, from physics to biology and chemistry to finance, economics or medicine and psychology. Independent from their roots in various applications, these PDE and their solutions frequently exhibit similar interesting structures that are investigated by mathematical methods from geometric analysis. Consequently, mathematical results in one direction may open doors to solving problems in entirely different fields or applications. One main research direction of the project concerns the Einstein equations from general relativity theory, which are the laws of the universe, linking its physical content to geometry. This theory is also crucial to make GPS work. Investigating these equations will increase our understanding of the universe both in the large as well as in smaller regions such as galaxies, binary neutron stars or binary black holes. Beyond that, this project will not only answer open questions in physics, but also create new ideas for physical models. When two massive objects like neutron stars or black holes merge, then gravitational waves are produced and travel from the source through the universe as ripples in spacetime. For the first time, such waves were observed in 2015 by Advanced LIGO (aLIGO), marking the beginning of a new era where information from distant regions of the universe is decoded directly from the universe itself (different from telescopes). Unraveling the new structures will rely on synergies between mathematics, astrophysics and physics. The project will build on the PI’s prior results to develop new methods to achieve these goals. The project will also have direct impact in a broader sense via teaching and outreach activities. The PI will train students and postdocs in these fields, and through broad outreach activities also the public including underrepresented groups. The PI will attend conferences to communicate the results. The PI will also make the results available via the internet and publications. In this project the PI will develop new mathematical methods to investigate the Einstein equations and other nonlinear PDE describing physical phenomena. The PI will investigate: (1) the Cauchy problem for the Einstein equations focusing on (I) spacetimes with radiation, and (II) the formation of black holes in GR when the Einstein equations are coupled to matter systems, (2) the mathematics of gravitational waves, their memory and related effects in GR as well as analogs of memory in other physical theories, (3) Euler equations and other PDE. These main directions comprise several projects. Many of them will rely on the PI's former results but also require new ideas and new mathematical methods. The PI's recent results for the Einstein equations in vacuum and with neutrino radiation revealed a panorama of new structures in gravitational radiation and memory that are expected to be seen in current and future gravitational wave detectors. Parts of the planned research link the mathematical insights to experiments (LIGO/VIRGO in particular). Moreover, the gravitational wave memory is expected to be detected in the near future. The PI and D. Garfinkle derived two analogs of memory within the electromagnetic theory. These are expected to be measured in an experiment as well. The mathematical methods are widely applicable to a broad spectrum of problems from mathematics to physics. The PI and collaborators will continue their research to complete the understanding of gravitational radiation and memory in GR, and to extend their research to other physical theories. The PI's research on the mathematical investigation of the Einstein equations in GR provides geometric-analytic methods to tackle other PDE. Moreover, newly found structures will be important in geometry as well as in PDE theory.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.