In these two talks I will discuss the recent proof by Brendle and Schoen of the differential 1/4 pinched sphere theorem and related recent work of Petersen and Tao. The differential 1/4-pinched sphere theorem states that if a simply connected compact Riemannian manifold has all its sectional curvatures pinched strictly between 1 and 4 then it is diffeomorphic to a sphere. The statement in the "homeomorphic" case is the classic sphere theorem of Berger and Klingenberg. Brendle and Schoen's proof involves studying the Ricci flow on manifolds with non-negative isotropic curvature. The first talk will consist of an introductory overview of the Ricci flow, including the recent fundamental work of Boehm and Willking, and a discussion of isotropic curvature, culminating in an outline of Brendle and Schoen's proof. In part II I will focus on two related results. Namely I will discuss the classification of manifolds with sectional curvatures only weakly quarter pinched, also due to Brendle and Schoen and work on the below one quarter pinched case due to Petersen and Tao.