Partial differential equations in conformal geometry

杰青博士提出研究保形几何偏微分方程领域的各种问题。其核心是流形（子流形）几何结构之间相互作用的研究和共形不变偏微分方程的分析，其一般原理源于数学物理中弦理论中所谓的AdS/CFT对应。目前有两种方法反映了这一原则。一类是利用散射算子对Yamabe问题进行推广的不同阶的Yamabe问题，另一类是利用双曲空间中的支持函数研究超曲面而得到的完全非线性Yamabe问题。本研究旨在研究量子引力理论与某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。这一建议的一个更广泛的影响是，该领域的研究进展将有助于建立理论物理学中提出的理论的数学框架。这一提议的第二个更广泛的影响是教育。这项研究将使Jie Qing博士能够积极参与加州大学圣克鲁兹分校的本科和研究生项目并做出贡献。

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.