Differential Equations and the Geometry of Manifolds

The focus of this project is to better understand the relationship between the geometry and the topology of a space. The latter, topology, is the study of properties of a space that are invariant under continuous stretching or bending of a space. Geometry involves understanding distances.

For example, if the surface of our planet is viewed as a sphere one can measure distances on it by computing arclengths of great circles. One can imagine deforming the Earth by pushing in or pulling on small or large regions to warp the geometry. There are many ways to make this notion precise in terms of minimizing a total energy measurement.

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 that use the least amount of energy, and maximize the symmetries of the space.

These projects will define appropriate energies on such spaces, and seek out the important optimal geometries that minimize the total energy. The PI will participate in mentoring, outreach and organization of conferences in the mathematics community.

In more technical terms, this research will 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 study of gravitational instantons in dimension four (both compact and complete non-compact), the study of collapsing sequences Ricci-flat metrics on K3 surfaces, the construction of a global moduli space of scalar-flat Kahler ALE metrics, and the study of the orbifold Yamabe problem.

In ongoing work with Hein, Sun, and Zhang, the PI has constructed new examples of Ricci-flat metrics on K3 surfaces, which collapse to an interval, with Heisenberg nilmanifolds occurring as fibers in the regular collapsing regions. In this case, gravitational instantons of type ALH-star bubble off, and it is of interest to have a better understanding of this class of instantons.

In joint work with Chen and Zhang, the PI has constructed examples of collapsing sequences of Ricci-flat metrics on the K3 surface that has both ALG and ALG-star bubbles, and it is also of interest to have a better understanding of these types of instantons, which have quadratic volume growth. In joint work with Han, the PI will conduct further study of the moduli space of scalar-flat Kahler ALE metrics for certain groups at infinity and finding new examples of such metrics on non-Artin components of deformations of isolated quotient singularities.

In joint work with Ju, the PI is studying compactness and existence results for the orbifold Yamabe problem, which differs substantially from the smooth case due to the failure of the positive mass theorem for ALE metrics. Finally, the PI is committed to integrating research and education and cultivating intellectual development on many levels.

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.