Differential Equations and the Geometry of Manifolds

University Of California-Irvine, Irvine CA

Investigators

Abstract

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 that 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 supported by this award are to define appropriate energies on such spaces, and to seek out the important optimal geometries which minimize the total energy. The PI is committed to integrating research and education and cultivating intellectual development on many levels, and plans to continue to be active in outreach and organization of conferences and other events in the mathematics community. 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 classification of gravitational instantons in dimension four, understanding Gromov-Hausdorff limits of Einstein metrics in the collapsing case, and the study of higher-dimensional complete non-compact Calabi-Yau structures. The PI has contributed to the classification of gravitational instantons in dimension 4, and plans to further investigate the global structure of the moduli spaces of these metrics. The PI has also contributed to the classification of asymptotically Calabi Calabi-Yau structures, and plans to study other types of complete non-compact Calabi-Yau structures in higher dimensions. The PI has contributed to the understanding of Gromov-Hausdorff limits of Calabi-Yau metrics on compact manifolds, and plans to further study global properties of compactifications of such moduli spaces. 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.