Geometric Partial Differential Equations and Complex Geometry

Northwestern University, Evanston IL

Investigators

Abstract

This project is concerned with the study of problems of geometric nature, often involving the curvature of a space or object, using primarily tools from partial differential equations. This is a central field in mathematics, which has ramifications and connections in physics and other sciences. One of the main themes of this research is the study of a class of spaces, known as Calabi-Yau, which play an important role in mathematics as well as high energy theoretical physics. According to string theory, our four-dimensional physical space-time possesses six extra dimensions which are extremely small, so that we don't normally perceive them, but are crucial for understanding elementary particles. These six dimensions together form a tiny Calabi-Yau space, which captures essential features of particle physics. Understanding its geometry would allow us to understand how particles are created and how they interact, and is one of the main current problems in mathematical physics. The PI will use techniques from geometric analysis and nonlinear partial differential equations to investigate problems about the geometry of complex and symplectic manifolds. The first project is about understanding limits of Ricci-flat Calabi-Yau manifolds as the Kahler class degenerates. This is closely related to the theory of mirror symmetry, which was inspired by physical considerations. The second project concerns the long-time behavior of the Ricci flow on compact Kahler manifolds, in the most difficult case when collapsing occurs at infinite time. The Ricci flow was used spectacularly to prove the Poincare and Geometrization conjectures for 3-manifolds, and understanding its behavior on higher-dimensional manifolds is a central problem in the field. The third project is centered on Donaldson's program to extend Yau's solution of the Calabi Conjecture in Kahler geometry to symplectic four-manifolds. This would provide a new analytic tool to construct symplectic forms four-manifolds as solutions of a highly nonlinear PDE, and would have striking applications in symplectic topology. 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.