Dispersive Partial Differential Equations in General Relativity

Most of my research is dedicated to the study geometric hyperbolic partial differential equations that appear naturally in General Relativity. In particular, one would like to understand the effect of complicated background geometry on the solution to wave and dispersive partial differential equations, and to apply the decay obtained for the linear equation to the nonlinear setting. I would like to understand the effects of phenomena such as the red shift effect and trapped null geodesics in black hole spacetimes have on global well-posedness results for nonlinear wave equations. Examples of such equations are semilinear wave equations with power nonlinearities, and quasilinear wave equations. In the long run, one of the goals of my work is understanding the stability of black holes under small perturbations of initial conditions. Recently I also became interested in the topic of cycle detection in discrete dynamical systems. The research is concerned with finding optimal ways to add a nonlinear delayed feedback control in order to stabilize cycles of dynamical systems, and uses tools from applied mathematics, harmonic analysis and complex analysis.