Ricci curvature, Analysis and Geometry

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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to be considered include how to effectively bring in the Ricci curvature information when the comparison theorems are no longer applicable; how to understand the interaction between the Ricci curvature and analysis on manifolds admitting extra structures; how to obtain sharp dependency on the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.