The aim of the course is to introduce characteristic classes of manifolds and singular varieties. The course will be elementary and will follow the historical development of the theory. Starting with the Poincaré-Hopf Theorem, we will introduce obstruction theory, which is the good theory for understanding the meaning of characteristic classes. Alternative definitions will be provided for complex and real manifolds (Chern and Stiefel-Whitney classes). In the vein of obstruction theory, we will explain the nice Marie-Hélène Schwartz proof of Poincaré-Hopf Theorem in the case of singular varieties. That construction provides her definition of characteristic classes for singular complex analytic varieties. In the algebraic case, the MacPherson definition of such classes, answering a Deligne and Grothendieck conjecture, is seen to be equivalent,leading to the so-called Schwartz-MacPherson classes.
We will discuss the different definitions of characteristic classes: Mather, Fulton and the Milnor classes. The course will conclude with more recent constructions of motivic classes and stringy classes.
This course will be planned in concondance with the course of José Seade.