The Curvature of 4-Manifolds

The Einstein equations of general relativity govern the gravitational field by constraining the curvature of a 4-dimensional space-time. This project concerns a family of generalizations of these equations in the Riemannian setting, where there is no difference between space and time, with an emphasis on trying to determine which 4-dimensional spaces carry solutions of these equations. The project will also help train a group of graduate students in mathematical research through their participation in activities related to the project. This project specifically seeks to discover and elucidate fundamental links between curvature and topology in dimension four. One focus is the construction of canonical metrics on compact manifolds, primarily by means of Kaehler and conformal geometry. Another focus is on understanding the relevant moduli spaces of solutions, and associated bubbling phenomena. Progress in this area could have significant repercussions outside of pure mathematics, especially because many of the geometrical questions to be investigated are intrinsically linked to problems in classical and quantum gravity, as well to other areas of theoretical physics. The project also includes training and mentoring of students and junior researchers. 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.