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 –spheres. For instance, when is a homotopy –sphere, the width is loosely speaking the area of the smallest –sphere needed to ‘pull over’ . Second, we use this to conclude that Hamilton’s Ricci flow becomes extinct in finite time on any homotopy –sphere.
"Width and finite extinction time of Ricci flow." Geom. Topol. 12 (5) 2537 - 2586, 2008. https://doi.org/10.2140/gt.2008.12.2537