Ricci solitons play an important role in the study of formation of singularities in the Ricci flow. They are self-similar solutions to the Ricci flow and natural extensions of Einstein metrics. In addition, (compact) Ricci solitons are critical points of certain geometric functional defined by Perelman. In this talk, I shall present some recent progress on classifications of shrinking Ricci solitons and stability of Ricci solitons with respect to Perelman's shrinker entropy.

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