Problems in Complex Geometry, Partial Differential Equations, and Mathematical Physics

The search for a unified theory of all forces of nature has led through string theories to new equations requiring a challenging and unfamiliar geometry. This research is devoted to the development of a theory of geometric partial differential equations which can address such equations. Such a theory is needed progress is to be made in this area of interface between partial differential equations, geometry, and high energy physics, and arrive some day at an understanding of physical laws at their most fundamental level. The training of both undergraduate and graduate students is an integral component of the project. It is particularly important for the next generation of scholars to be introduced to these difficult issues as soon as possible.

A characteristic feature of these new equations is a curvature condition combined with a cohomological constraint. Thus they can be viewed as defining “canonical metrics”. The PI will strive to develop geometric flow methods to find such metrics, and to consider flows of both tensor and spinor fields. These geometric flows are interesting in their own right, from the point of view of the theory of partial differential equations. Of particular interest is to determine the manifestations of supersymmetry and duality, which are expected in particular to link aspects of complex geometry to aspects of symplectic geometry. A hybrid cohomology theory will also be developed with possible applications to superstring perturbation theory.

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.