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

爱因斯坦方程和其他偏微分方程的几何分析研究

基本信息

批准号：
1811819
负责人：
Lydia Bieri
金额：
$ 21.14万
依托单位：
Regents of the University of Michigan - Ann Arbor
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-09-01 至 2022-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811819&HistoricalAwards=false
关键词：
Geometric Analytic Studies Einstein Equations

项目摘要

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.

该项目涉及数学和物理以及这些领域之间的相互作用的一些主题。这些主题的首要目标是推进我们在数学和物理学方面的知识。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。该项目的新见解和方法对于解决其他结构相似的PDE也很重要，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涉及爱因斯坦方程，该方程将我们宇宙的物理内容与几何学联系起来，从而赋予物理定律一种几何结构。这些方程是广义相对论的核心，它支配着我们宇宙的物理学。探索这些方程将导致更好地了解整个宇宙和孤立的系统，如星系，双黑洞或双中子星。 2015年，先进的LIGO（aLIGO）项目对引力波的观测标志着一个新时代的开始，在这个新时代中，来自宇宙遥远区域的信息直接从宇宙本身解码（而不是从电磁波，例如望远镜中的光）。现在比以往任何时候都更需要数学，特别是偏微分方程和几何、物理和天体物理学之间的协同作用来解开新的结构。PI将在她的研究结果的基础上开发新的方法来实现这些目标。通过该项目的教育部分，PI的研究还将通过教学和推广活动产生更广泛的直接影响，PI将培训这些领域的学生和博士后，并通过广泛的推广活动接触公众成员，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研究结果。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。特别是，PI建议研究：（1）爱因斯坦方程的柯西问题，重点是辐射时空;（2）引力波的数学及其记忆效应（时空的永久变化，表现为像aLIGO这样的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;（3）Euler方程和其它偏微分方程本身及其与Einstein方程的耦合。虽然所有这些主题都围绕着纯粹的数学（几何分析）处理，但它们中的每一个都将产生可以直接应用于物理实验的结果。部分建议的研究将继续PI以前的工作，将她的数学见解与实验（特别是aLIGO）联系起来。预计在不久的将来，引力波记忆效应将被测量。这些新项目将部分建立在PI和合作者最近的成果和方法的基础上，但也需要新的想法和方法。PI和D。加芬克尔在电磁学中推导出了两个类似的记忆，这是第一次在广义相对论之外。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整体柯西问题的其他项目联系起来，她将用几何分析方法研究其他偏微分方程。此外，爱因斯坦方程的解决方案耦合到其他偏微分方程将被调查的一般情况下，包括渐近平坦以及宇宙时空，从而有趣的本地和全球的结构，预计将被发现。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。