Ricci Flow and Geometric Applications : Cetraro Italy 2010 Preface The Differentiable Sphere Theorem (after S. Brendle and R. Schoen) Thick Thin Decomposition of three–manifolds and the Geometrisation Conjecture Singularities of three–dimensional Ricci flows Notes on K¨ahler Ricci flow. Presenting some impressive recent achievements in differential geometry and topology this volume focuses on results obtained using techniques based on Ricci flow. These ideas are at the core of the study of differentiable manifolds. Several very important open problems and conjectures come from this area and the techniques described herein are used to face and solve some of them. The book’s four chapters are based on lectures given by leading researchers in the field of geometric analysis and low dimensional geometry topology respectively offering an introduction to: the differentiable sphere theorem (G. Besson) the geometrization of 3 manifolds (M. Boileau) the singularities of 3 dimensional Ricci flows (C. Sinestrari) and Kähler–Ricci flow (G. Tian). The lectures will be particularly valuable to young researchers interested in differential manifolds.