An important motivation for the research of the PI is to understand the relationship between the geometry and the topology of a space. The latter, topology, is the study of properties of a space which are invariant under continuous stretching or bendings of a space, while the former, geometry, involves understanding distances and is more rigid. For example, the surface of our planet is a sphere, and one measures distances on it by computing arclengths of great circles (the Earth is actually an oblate spheroid, but it is very close to being perfectly spherical). One can imagine deforming the Earth by pushing in or pulling on small or large regions to warp the geometry. Such a deformation is less appealing than the familiar round Earth, and there are many ways to make this notion very precise in terms of minimizing some sort of total energy measurement. This is directly related to physical principles which say that the state of a physical system will tend towards a final configuration which minimizes the total energy. This idea can be generalized to higher-dimensional objects called manifolds, which are generalized versions of the surface of our planet. For example, the space that we live in is three-dimensional, and if one includes time, we are in a four-dimensional universe. In order to understand these types of higher-dimensional objects, one attempts to find the best way to measure distances on them which use the least amount of energy, and maximize the symmetries of the space. The projects in this proposal are to define appropriate energies on such spaces, and to seek out the important optimal geometries which minimize the total energy.
In more technical terms, the research of the PI is, broadly speaking, to use solutions of partial differential equations which are geometric in origin to study properties of differentiable manifolds. The main areas of concentration of the PI's research are the existence of critical metrics generalizing the Einstein condition, the study of quadratic curvature functionals on Riemannian manifolds, the study of critical ALE metrics and orbifolds, and properties of moduli spaces of critical metrics. In joint work with Matt Gursky, the PI has proved existence of critical metrics on various four-manifolds. The proposed research is to further study the properties of the moduli space of such solutions. This is related to orbifold compactness theorems previously studied by Tian-Viaclovsky. The PI has previously demonstrated non-solvability of the Yamabe problem on certain compact orbifolds, which showed that the orbifold Yamabe problem is more subtle than in the case of smooth manifolds. The PI proposes further exploration of this phenomenon, and of connections with the notion of mass of ALE spaces. There has been a considerable amount of research on the existence of anti-self-dual metrics on compact manifolds; they have been shown to exist in abundance. Another goal of the proposal is therefore to understand global properties of the moduli space in certain cases; especially for orbifold-cone anti-self-dual metrics. Finally, the PI is committed to integrating research and education and cultivating intellectual development on many levels. The PI has been active in outreach and organization of conferences in the mathematics community.
