Fundamental groups in Algebraic Geometry 1000-1M24GPA

The fundamental group of a topological space is one of the basic homotopy invariants. It allows one to translate questions about the "shape" of the space into group-theoretical considerations. As it turns out, similar invariants can be introduced for algebraic varieties defined over arbitrary fields and other algebra-geometric objects. During the course we shall learn some basic facts about fundamental groups of complex algebraic varieties and define the etale fundamental group. This invariant, introduced by Grothendieck, allows one in particular to interpret the Galois group as an example of a fundamental group, providing a foundation for modern arithmetic geometry.

The goal of the course is to cover several variants of the notion of a fundamental group in algebraic geometry as well as methods of their study. We will assume basic familiarity with schemes.

Tentative outline:

0. Fundamental groups in topology – review and reinterpretation
I. Fundamental groups of complex algebraic varieties and Kaehler manifolds
II. The etale fundamental group

Additional topics:
a. Anabelian geometry
b. The pro-etale fundamental group of Bhatt and Scholze
c. Tannakian fundamental groups and differential Galois theory