PhD course: Differential Topology

Differential topology is the study of manifolds, focusing on their global, topological properties, rather than local, analytic ones. The subject breaks into sub-areas: three-dimensional, four-dimensional, and “high”-dimensional manifolds, which usually means dimensions 5 and above. In this course we will mostly focus to the latter. The goal is to learn the major techniques for working with high-dimensional manifolds: the Whitney trick, surgery, cobordism, and some of their classic applications: the h-cobordism theorem, the high-dimensional Poincare conjecture, exotic smooth structures on high-dimensional spheres. Depending on time and interest, I hope to also make excursions into more recent developments, such as embedding calculus and the stable homology of mapping class groups.