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

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.