Symmetry and geometric partial differential equations

这个项目旨在开发工具来帮助研究偏微分方程式，偏微分方程式是我们理解物理世界的基础。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，欧几里得空间和球面，这种对应才能被很好地理解。使用共形几何学中的强大几何工具，该项目将把这一点扩展到不那么对称的空间。该项目产生的知识将扩展到更一般的几何背景，为研究弯曲空间中相关的自然方程提供了具体的背景。

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.