Classical Galois theory, whose original purpose was to give a reasonable necessary and sufficient condition for the solvability in radicals of an algebraic equation, has been modified and widely generalised many times. The list of important examples of Galois/covering theories includes: theory of covering spaces, which goes back to H. Poincaré and is often considered as the main origin of algebraic topology; étale Galois theory of schemes, due to A. Grothendieck, which plays an important role in algebraic geometry and has important applications in algebraic number theory; many others, involving several areas of abstract algebra and geometry/topology.

This course will describe the current stage of the generalisation process, including the so-called categorical Galois theory, which is supposed to unify and simplify all known cases, and most recent noncommutative versions of Galois-theoretic constructions that are still to be understood categorically. Among them are Hopf-Galois extensions, coalgebra-Galois extensions, Galois corings and comodules, and principal actions of quantum groups on C*-algebras (noncommutative principal bundles). The course is divided into three parts:

Part 1: The first goal is to give an overview of structures appearing in geometrically significant generalisations of classical Galois theory. We will focus on the following topics: Galois field extensions; Galois theory as equivalence of categories; fundamental group and universal covering; torsors and Galois cohomology; Galois context and principal fibrations; Galois context in monoidal categories; Tannakian categories.

Part 2: This part will be devoted to Galois theory in general categories and some of its examples, with necessary preliminary material from category theory included: universal properties, Yoneda embedding, and discrete fibrations; adjoint functors, monads, and algebras over monads; internal category theory and Grothendieck descent; Galois theory in general categories; Galois theory of abstract families in algebra and geometry; other examples of Galois theories.

Part 3: The final part will be devoted to noncommutative Galois theory, specifically to the following recent developments: Hopf-Galois extensions: crossed products as cleft extensions, Schneider's theorems; differential-geometric aspects of Hopf-Galois theory (strong connections, principal comodule algebras, free actions of compact quantum groups on unital C*-algebras); the Chern-Galois character (entwining structures, principal extensions, associated modules, index pairings); corings and comodules (the Sweedler coring and the coring associated to an entwining structure, comonads and associated adjoint functors, separable and split algebra extensions, corings with a grouplike element and their relation to differential graded algebras and connections); Galois comodules (corings associated to finitely generated projective modules and beyond, comatrix contexts, Galois corings, pre-dual Bourbaki-Jacobson correspondence, "noncommutative descent", i.e., Galois condition and monadicity).

Prerequisites: The course is essentially self-contained. Only basic experience and familiarity with simplest properties of rings, modules, groups, Hopf algebras, topological spaces, fibrations, and categories is expected. No particular knowledge about Galois theory is assumed.