Conformal geometry and partial differential equations

杰青博士主要研究共形几何中的各种问题和相关的偏微分方程。散射矩阵研究的最新进展为理解物理学中的全息原理提供了一个较为全面的途径，这是非常令人着迷和有希望的。在这个项目中，Jie Qing博士提出研究全息原理的数学基础的几个方面，特别是所谓的AdS/CFT对应。杰青博士还建议继续研究广义相对论数学中出现的几个几何问题。这一建议的智力价值在于，这些研究的结果将更好地理解低维流形的几何结构。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。这一提议的一个更广泛的影响是，这一研究领域的进步将极大地提高我们对自然理论的理解。拟议的研究项目将研究活动作为加州大学圣克鲁斯分校数学系本科和研究生教育计划的一部分。

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.