This page presents my activity and plans supported by the NSF Award #2309084.
The goal of the award is to investigate the role of hyperbolic geometry in relativity (and related areas) with a focus on the computation of gravitational waves from astrophysical sources.
This is a fairly ambitious project, as the reviewers have stated. The research problems include very distinct areas such as black-hole perturbations, nonlinear Einstein equations, applied mathematics for engineering applications, and even quantum gravity.
The common thread is the use of hyperbolic geometry to model the space extending from a source of radiation to a far away observer. The hyperbolic nature of this space is a consequence of the finite speed of light. Incorporating this property of space in numerical computations has resulted in dramatic improvements over the past 15 years. In this award, I will combine mathematical analysis, numerical experiments, and physical insight to explore the applications of hyperbolic geometry in diverse areas of physics and technology.
I divide my activities for this award into three categories.