Partial differential equations in conformal geometry

共形几何中的偏微分方程

• 批准号: 1303543
• 负责人: Jie Qing
• 金额: $ 20.02万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1303543&HistoricalAwards=false
• 关键词: Partial; differential; equations; conformal; geometry

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.

杰青博士提议研究共形几何偏微分方程领域的各种问题。其核心是研究流形（子流形）几何结构之间的相互作用和共形不变偏微分方程的分析，其一般原理源自数学物理学中弦论中所谓的AdS/CFT对应关系。 目前有两种方法体现了这一原则。一种是通过散射算子推广 Yamabe 问题的不同阶的 Yamabe 问题，另一种是通过双曲空间中的支持函数研究超曲面而产生的完全非线性 Yamabe 问题。拟议的研究是研究与量子引力理论和一些共形场论相关的全息原理。它已成为数学家和物理学家互动领域的一部分。该提案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。该提案的第二个更广泛的影响是对教育的影响。拟议的研究将使杰青博士能够积极参与加州大学圣克鲁斯分校的本科和研究生项目并为其做出贡献。