CAREER: Geometric-Analytic Investigations of Spacetimes and their Nonlinear Phenomena

这个项目涉及研究非线性偏微分方程组，即广义相对论中的爱因斯坦方程，解的时空几何及其在物理学和天体物理学中的意义。这项研究的主要目的之一是为引力波永久移动试件质量的非线性记忆效应推导出一个完整的理论。这将包括回答在双中子星合并中是否存在中微子辐射的这种效应，以及是否存在宇宙学记忆效应的问题。目标是在各种情况下给出详细的几何分析描述，并将数学应用到实验中。在过去的几年里，国际和平研究所已经在这个方向上做出了贡献。该项目正在出现的结果预计将回答有关引力辐射、天体物理学中的源物体以及我们的宇宙非常早期的基本物理问题。自从D.Christodoulou在他的开创性文章中推导出非线性记忆效应(也称为Christodoulou效应)以来，电磁场与爱因斯坦方程的耦合是否影响这种非线性现象一直是一个突出的问题。PI与合作者P.Chen和S.T.Yau解决了这个问题，证明了爱因斯坦-麦克斯韦(EM)方程中的电磁场扩大了Christodoulou效应。利用Christodoulou对爱因斯坦真空方程的几何解析方法和N.Zipser对EM方程的分析，Pi、Chen和Yau在EM方程的几何分析中建立了新的结果和方法。Christodoulou以及PI与Chen和Yau进行的几何解析调查使作者能够推导出精确地描述所有数据的相关效应的精确解。特别是，这一结果适用于大型数据，如双星黑洞或双星中子星合并。自那以后，几何分析已被证明是解决这些问题的最有力的方法。在研究引力波的非线性记忆效应时，洛伦兹几何中的零点超曲面起着至关重要的作用。这是因为引力波沿着这样的零超曲面传播，而实验是在零无穷远进行的。因此，必须了解后者的渐近行为。克里斯托杜卢和克莱纳曼在他们的著作《明可夫斯基空间的全球非线性稳定性》中开发了新的技术来研究这些问题。PI推广了他们的工作，并从数据衰减的角度建立了边界情况。在一本300页的研究专著中，PI获得了许多新的结果，这些结果可以应用于几何和分析的其他问题。最近，主要方法及其发展导致了数学物理中一些最具挑战性的问题的解决。这个新项目将使用PI最近工作中的新想法，预计它将在广义相对论(GR)、几何学和偏微分方程分析方面产生进一步的新结果。该提案的教育部分侧重于向公众传播数学、物理和天体物理学知识以及历史背景。该教育项目的目标是在少年派的家乡大学自然历史博物馆创造一个将几何和分析与天体物理学相结合的展览，该教育项目将包括本科生和研究生的学生工作。他们将与PI和博物馆工作人员一起参与该项目的高中教学工作。在博物馆举办的公开活动中，国际和平协会将与博物馆工作人员合作，介绍促进学习的活动。研究成果将以简化的方式呈现，并适用于观众。该项目旨在接触到非常广泛的公众，包括少数群体、有小孩的家庭、各级学校，并将对每个人都有吸引力。一些长期的影响将是高中课堂上通过引入物理与历史和艺术相结合的概念来教授数学的新活动。广义相对论的几何本质使几何分析成为回答许多物理问题的完美研究领域。GR定律是爱因斯坦方程，它将时空的曲率与其物质含量联系起来。探索这些方程式将使我们更好地理解整个宇宙和孤立的系统，如星系、双星黑洞或双星中子星。解开几何、分析和物理之间复杂的相互作用是GR的主要目标之一。这个项目旨在研究这种美丽的相互作用。在这一过程中，数学工具的发展有望在许多结构相似的非线性偏微分方程组的分析中取得成果。通过教育部分，国际和平研究所的研究也将通过上述活动在更广泛的意义上产生直接影响。私人助理将通过出席会议交流她的成果来补充外联工作。她还将通过互联网和出版物公布她的研究结果。

This project concerns the investigation of nonlinear partial differential equations (PDEs), namely the Einstein equations in general relativity, the geometry of the solution spacetimes and their implications in physics and astrophysics. One of the main aims of the proposed research is to derive a complete theory for the nonlinear memory effect of gravitational waves displacing test masses permanently. This will include answering the questions whether there exists such an effect for neutrino radiation as present in binary neutron star mergers and if there is a cosmological memory effect. The goal is to give a detailed geometric-analytic description in the various scenarios and to apply the mathematics to experiments. The PI has already made contributions in this direction in the recent past. The emerging results of the project are expected to answer fundamental physical questions about gravitational radiation, the source objects in astrophysics and the very early period of our universe. Since the pioneering article by D. Christodoulou, where he derived the nonlinear memory effect (also called the Christodoulou effect), it had been an outstanding problem if electromagnetic fields coupled to the Einstein equations affect this nonlinear phenomenon. The PI with collaborators P. Chen and S.-T. Yau solved the problem, proving that the electromagnetic field in the Einstein-Maxwell (EM) equations enlarges the Christodoulou effect. Working with Christodoulou's geometric-analytic approach for the Einstein vacuum equations and N. Zipser's analysis of the EM equations, the PI, Chen and Yau established novel results and methods in the geometric analysis of the EM equations. The geometric-analytic investigations by Christodoulou as well as by the PI with Chen and Yau allowed the authors to deduce exact solutions describing the related effects precisely and for all data. In particular, the results hold for large data such as binary black hole or binary neutron star mergers. Geometric analysis has since proven to be the most powerful method to tackle these problems. In the study of the nonlinear memory effect of gravitational waves, null hypersurfaces in Lorentzian geometry play a crucial role. This is due to the fact that gravitational waves travel along such null hypersurfaces and experiments are performed at null infinity. Therefore, the asymptotic behavior of the latter has to be understood. D. Christodoulou and S. Klainerman in their work "The global nonlinear stability of the Minkowski space" developed new techniques to investigate those. The PI generalized their work and established the borderline case from the point of view of decay of the data. In a research monograph of 300 pages the PI obtained numerous new results that can be applied to other problems in geometry and analysis. The main method and its developments have recently led to the solution of some of the most challenging problems in mathematical physics. The new project will use new ideas from the PI's recent work and it is expected to yield further new results in general relativity (GR), geometry and PDE analysis. The educational component of the proposal focuses on disseminating knowledge of mathematics, physics and astrophysics as well as historic background to the public. Aiming at the creation of an exhibit at the Museum of Natural History at the PI's home University combining geometry and analysis with astrophysics, the educational project will include student work at undergraduate and graduate levels. Together with the PI and the museum staff, they will be involved in teaching at high schools working on the project. At public events organized by the museum the PI will collaborate with the museum staff to present activities to enhance learning. Research results will be presented in a simplified way and adapted to the audience. The project is designed as to reach out to a very broad public including minority groups, families with small kids, schools at all levels and will have an appealing component for everyone. Some of the long-lasting impacts will be new activities for high school classes to teach mathematics by introducing concepts of physics combined with history and art. The geometric nature of general relativity singles out geometric analysis to be the perfect research field to answer the many physical questions. The laws of GR are the Einstein equations, linking the curvature of spacetime to its matter content. 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. To unravel the complex interplay between geometry, analysis and physics is one of the main goals in GR. This project aims to investigate this beautiful interaction. Along the way, the mathematical tools developed promise to bear fruit in the analysis of the many structurally similar nonlinear PDEs. Through the educational component, the PI's research will also have direct impact in a broader sense via the afore-mentioned activities. The PI will complement the outreach by attending conferences to communicate her results. She will also make her results available via internet and publications.