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.
DMS 1005295. I used it to support my participation to the research program on conformal geometry and geometric PDE June 17 - July 7 2013 at CNR, Barcelona, Spain. I delivered a research talk on scalar invariants of surfaces in conformal 3-sphere. We all know that the seminal work of Fefferman and Graham on local scalar invariant of conformal geometry becomes the fundamental building block for the current study of conformal geometry including the mathematics for teh so-called AdS/CFT correspondence proposed in the quantum theory of graviaty in theoretic physics. In this talk I repoted our project to initiate the study of local scalar invariant for submanifolds in a conformal manifold. I also used this grant to support my student Wei Yuan and myself to attend the summer school of PCMI 2013 on the geometric analysis and mathematical general relativity, Jun 24 - July 20, at Park City, UT, where he and I continued to work on our research project on scalar curvature rigidity of static spaces. The positive mass theorem is a very influential work in mathematical relativity. The positive theorem has different versions modelled on Euclidean space and hyperbilic space. There has been attempt to extend it to cover spherical case known as the Minn-Oo conjecture. We would like to first to extend local scalar curvature rigidity on static spaces in general. It is known that local scalar curvature rigidity is weaker than global scalar curvature rigidity, while positive mass theorem is even stronger than global scalar curvature rigidity.
