Conformal geometry and partial differential equations

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Dr. Jie Qing proposes to study various problems in conformal geometry and related partial differential equations. Recent development in the study of the scattering matrix in providing rather global way to understand the holography principle in physics seems very fascinating and promising. Dr. Jie Qing proposes in this project to investigate several aspects of mathematical foundations of the holography principle, particularly the so-called AdS/CFT correspondence. Dr. Jie Qing also proposes to continue his study of several geometric problems arising from the mathematics of general relativity. The intellectual merit of this proposal is that consequences of these investigations will give better understanding of geometric structure of manifolds of low dimensions.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 greatly improve our understanding of the nature in theory. The proposed research project incorporates research activities as a part of the undergraduate and graduate education programs in the department of mathematics at UC Santa Cruz.