Curvature and topology of manifolds

The interplay between curvature and topology of Riemannian manifolds is among the most fundamental questions in differential geometry. Over the past century, various different approaches have been developed to attack these types of problems. This includes variational techniques based on geodesics and minimal surfaces, as well as the Ricci flow approach pioneered by Richard Hamilton. In this lecture, I will give an overview of the subject, focusing on the positive curvature setting.