Characteristic classes

Characteristic classes are one of the main objects in algebraic topology. The theory of characteristic classes had been initiated in works by Steifel, Whitney, Pontryagin, Chern and many others in 30–40s of the last century. Since that time it had been intensively developed and had become a necessary tool in most of branches of the modern topology. There are various approaches to constructing characteristic classes. We will give their overview and explain relations between them. Besides, we will describe a number of applications including classification of real division algebras and existence of smooth structures on a 7-dimensional sphere. Also, we shortly discuss homotopical, topological, and combinatorial invariance (or non-invariance) of various characteristic classes. Basically, we will focus on classical characteristic classes: Stiefel-Whitney, Euler, Pontryagin, and Chern classes. If time permits, we will also discuss more recent objects such as Mumford-Miller-Morita classes of fibrations by surfaces.
The talk is for non-topologists, we provide all necessary preliminaries and definitions.