Partial differential equations in conformal geometry

杰青博士建议研究保角几何中偏微分方程组领域的各种问题。它的核心是研究流形(子流形)的几何结构和共形不变偏微分方程的分析之间的相互作用，其一般原理源于数学物理中弦理论中的所谓ADS/CFT对应。目前有两种方法反映了这一原则。一个是通过散射算子推广Yamabe问题的不同阶Yamabe问题，另一个是通过双曲空间中的支撑函数研究超曲面所产生的完全非线性Yamabe问题。这项研究是为了研究量子引力理论和一些共形场理论之间的全息原理。它已经成为数学家和物理学家可以互动的领域的一部分。这一提议的一个更广泛的影响是，这一研究领域的进步将有助于为理论物理中提出的理论建立数学框架。这项建议的第二个更广泛的影响是对教育。拟议的研究将使杰青博士能够积极参与并为加州大学圣克鲁斯分校的本科生和研究生项目做出贡献。

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.