Symmetry and geometric partial differential equations

该项目旨在开发工具，以帮助偏微分方程的研究，这是我们对物理世界的理解的基础。拉普拉斯方程的对称性是寻找和解释其解的基础，并且可以追溯到底层空间的共形对称性。只有对于最对称的空间，即欧几里得空间和球面，才能很好地理解这种对应关系。使用来自共形几何的强大几何工具，该项目将扩展到不太对称的空间。从这个项目产生的知识将扩展到更一般的几何背景下提供一个具体的设置为相关的自然方程在弯曲空间的研究。

This project aims to develop tools to assist the study of partial differential equations, which are fundamental to our understanding of the physical world. Symmetries of the Laplace equation are fundamental in both finding and interpreting its solutions and can be traced to the conformal symmetries of the underlying space. Only for the most symmetric of spaces, Euclidean space and the sphere, is this correspondence well understood. Using powerful geometric tools from conformal geometry, the project will extend this to less symmetric spaces. The knowledge generated from this project will extend to more general geometric contexts providing a concrete setting for the study of the associated natural equations in curved spaces.