Partial differential equations in conformal geometry

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

Dr. Jie Qing proposes to study various problems in the field of partial differential equations in conformal geometry. At its core is the study of the interplays between geometric structures of manifolds (submanifolds) and analysis of conformally invariant partial differential equations, with a general principle that is originated from the so-called AdS/CFT correspondence in string theory in mathematical physics. Currently there are two approaches that reflect such principle. One is the Yamabe problems of different orders generalizing the Yamabe problem via scattering operators, and the other is the fully nonlinear Yamabe problems arising from the study of hypersurfaces via support functions in hyperbolic space. The proposed research is to study the holography principles that relate quantum gravitation theory and some conformal field theory. It has become a part the field where mathematicians and physicists can interact. One broader impact of this proposal is that advancements in this field of research will help establish mathematical framework for the proposed theory in theoretic physics. The second broader impact of this proposal is to the education. The proposed research will enable Dr. Jie Qing to actively participate and contribute to the undergraduate and graduate program at UC Santa Cruz.