Associate Professor, Director Michigan Center for Applied and Interdisciplinary Mathematics (MCAIM)

### About

My research interests are primarily in general relativity theory, nonlinear partial differential equations, geometric analysis and differential geometry. More specifically, my research focuses on the Einstein equations in general relativity (GR). These equations govern the geometry of spacetime and thereby the phenomenon of gravitation. They constitute the physical laws of our universe. In particular, they exhibit a rich geometric structure that one has to reveal in order to answer questions from physics. GR is a rich source of many nonlinear hyperbolic and other problems in pde.

Among the many questions one would like to know under what conditions are galaxies and other astrophysical objects stable? When do singularities like black holes occur and what are the structures of these? I have been investigating global solutions to the Einstein equations and have studied the geometry of the solution spacetimes. A major goal in general relativity is to describe exactly spacetimes which do not collapse and form black holes. I have examined the geometry of spacetimes which are global solutions of the Einstein vacuum equations. I have studied, which type of initial data prescribed on a 3-dimensional spacelike manifold yields a complete, unique, global solution, namely a 4-dimensional spacetime manifold, with physically interesting properties. Thereby I have established sharp conditions on the decay of the data for the nonlinear stability to hold. Another major goal of general relativity and astrophysics is to precisely describe and finally observe gravitational radiation, one of the predictions of GR. Together with my collaborators I have derived the nonlinear electromagnetic Christodoulou memory effect of gravitational waves. The latter exhibit a nonlinear memory displacing test masses permanently. This nonlinear effect is named after Demetrios Christodoulou who found it in 1991 in the Einstein vacuum case. I have been working on many aspects of the Einstein equations. Further, I am very interested in problems arising in the analysis of other nonlinear hyperbolic partial differential equations. My research investigates the rich interplay of mathematics with physics and astrophysics.

Among my publications there are two books. One is a research monograph extending the stability result of Minkwoski space in general relativity for the Einstein vacuum equations, published with the American Mathematical Society - International Press. And the other book is co-authored with an astrophysicist explaining the discovery of the expanding universe, written for a broader scientific audience and published by Cambridge University Press.