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We start with some basic concepts in Riemannian geometry and discuss an important tool in the subject called the Ricci flow, which was introduced by Hamilton in 1982. Ricci flow can be viewed as a heat flow of the Riemannian metric which evolves in the direction of its Ricci curvature. Ricci soliton is a special solution to the Ricci flow and plays a crucial role in the singularity analysis of the flow. In the later part of the talk, we will cover some results which are based on my joint works with Man-Chun Lee and Luke Peachey.

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