# There are finitely many surgeries in Perelman's Ricci flow

# There are finitely many surgeries in Perelman's Ricci flow

Although the Ricci flow with surgery has been used by Perelman to solve the Poincaré and Geometrization Conjectures, some of its basic properties are still unknown. For example it has been an open question whether the surgeries eventually stop to occur (i.e. whether there are finitely many surgeries) and whether the full geometric decomposition of the underlying manifold is exhibited by the flow as $t \to \infty$. In this talk I will show that the number of surgeries is indeed finite and that the curvature is globally bounded by $C t^{-1}$ for large $t$. Using this curvature bound it is possible to give a more precise picture of the long-time behavior of the flow. The proof of this result builds on a previous theorem in which I established the finiteness of the surgeries as well as the curvature bound under a certain topological assumption. I will give a very brief overview of the proof of this theorem and explain why this extra assumption was crucial. Next, I will present new tools which enabled me to remove this assumption completely.