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

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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.