Topology of Differentiable Manifolds

MAT 3071

Topology was born in the early 20th century as a kind of “qualitative geometry,” and has become instrumental in dynamical systems, analysis, geometry, and algebra. Currently, topological methods are actively developed in applied areas, such as data analysis, machine learning, and physics. This course is an introduction to the topology of differentiable manifolds, in which the subject is approached from the viewpoint of intersection theory and Thom’s transversality theorem. We begin with a discussion of general (point-set) topology, abstract and embedded manifolds, as well as a discussion of Sard’s theorem, transversality, and stability. Next follows the introduction of mod -2 intersection numbers, together with some classical applications. We conclude with the study of oriented manifolds and oriented intersection theory, the Lefschetz fixed-point theorem, and the Poincar´e–Hopf index theorem, as well as the fundamental theorem of algebra. If time permits, we also discuss briefly differential forms on manifolds and de Rham cohomology. This course is accompanied by a weekly seminar/tutorial.