Martin Bacques: Hamilton's Ricci flow and three manifolds with positive Ricci curvature

In 1982, Richard Hamilton introduced a new evolution equation, the so-called Ricci flow, which was later used to prove many important results in riemannian geometry. In the same paper, Hamilton proved the following theorem: every 3-manifold with positive Ricci curvature admits a metric of constant sectional curvature. In this talk, I will present some of the main ideas behind the proof of this result. The argument uses the maximum principle together with derivative estimates for the curvature tensor to study the evolution of geometric quantities under the flow and to prove convergence to a smooth limit metric of constant sectional curvature.