Huai-Dong Cao of the Institute for Advanced Study gives a seminar talk on Ricci solitons, the self-similar solutions to Ricci flow that Richard Hamilton introduced in the mid-1980s as generalizations of Einstein manifolds. Cao opens with a short introduction to gradient Ricci solitons before turning to the shrinking case, which models Type I singularities in Ricci flow and arises as critical points of Perelman's nu-entropy functional. The core of the talk works through the linear stability of shrinking Ricci solitons with respect to that entropy, and derives first eigenvalue estimates for the Laplace-Beltrami operator and related Lichnerowicz-type Laplacian operators. Delivered at Simonyi Hall as part of the IAS Analysis and Mathematical Physics seminar series, the talk assumes graduate-level familiarity with Riemannian geometry and geometric flows, and is aimed at researchers working in geometric analysis rather than a general audience.