Width and finite extinction time of Ricci flow

This is an expository article with complete proofs intended for a general nonspecialist audience. The results are two-fold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal 2-spheres. For instance, when M is a homotopy 3-sphere, the width is loosely speaking the area of the smallest 2 sphere needed to 'pull over' M. Second, we use this to conclude that Hamilton's Ricci flow becomes extinct in finite time on any homotopy 3-sphere.