New Challenges in Geometric Wave Equations

The research project is concerned with novel mathematical questions for nonlinear geometric wave equations. This class of equations describes many aspects of modern physics, ranging from the simple vibrating string to general relativity and particle physics. In general, physical systems are characterized by an initial state whose time evolution is then obtained by solving the corresponding equation of motion. From the mathematical perspective, this requires a suitable solution theory for the equation. The development of such a theory for many different models is the main goal of the present research project. In particular, we will consider so-called control problems, where along with the initial state also the end state is prescribed and the latter should be obtained by applying external forces to the system. The natural question is then whether every end state can be attained and what the corresponding cost is. Furthermore, physical systems are always subject to perturbations and a mathematically admissible solution is physically relevant only if it is sufficiently stable against external influence. This requires the development of a suitable stability theory which will also be at the center of the research project. In particular, random perturbations will be studied. All of these questions are largely open for geometric wave equations and will be approached for the first time.