Mathematics MA, Differential Topology

The course provides an introduction to differential topology, the study of smooth manifolds through topological methods. A central theme is the analysis of mappings between manifolds and the properties preserved under diffeomorphisms. The course emphasizes how local phenomena can be connected to global structures, for example through Sard’s theorem and transversality. Fundamental results such as the Jordan–Brouwer separation theorem, the Borsuk–Ulam theorem, and classical connections between vector fields, fixed points, and curvature (including the Poincaré–Hopf, Lefschetz, and Gauss–Bonnet theorems) illustrate how differential topology unifies analysis, geometry, and topology.