Ricci curvature, Analysis and Geometry

本项目的基本主题是更深入地了解Ricci曲率在几何、拓扑和分析中的作用。需要考虑的问题包括：当比较定理不再适用时，如何有效地引入Ricci曲率信息；如何理解Ricci曲率与允许额外结构的流形分析之间的相互作用；如何获得各种几何偏微分方程组的解对Ricci曲率的强烈依赖性。Ricci曲率作为一种度量空间曲线如何从几何和物理中自然产生的方法。了解它对空间整体形状的影响一直是几何学中的一个重要课题。这个项目是关于通过分析方法从不同的角度观察利玛窦的曲率，从而获得新的见解。

The underlying theme of this project is to understand in more depth the role of Ricci curvature in geometry, topology and analysis. The problems to be considered include how to effectively bring in the Ricci curvature information when the comparison theorems are no longer applicable; how to understand the interaction between the Ricci curvature and analysis on manifolds admitting extra structures; how to obtain sharp dependency on the Ricci curvature of the solutions to various geometric PDEs.Ricci curvature, as a way of measuring how space curves, arises naturally from geometry and physics. It has always been an important subject in geometry to understand its effect on global shapes of the space. This project is about looking at, hence gaining new insight into, the Ricci curvature from different angles via analytical methods.