Hilma af Klint Primordial Chaos, No. 16 (1906)
Time: Wednesdays, 12:15-13:45 and 14:00–15:30
Venue: 3140 and 3170 at MIMUW
Office hours: Mondays, 10-11:30, 606 IMPAN
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.
Lecture notes: [PDF]
Problem sets: 1 (due Mar 13), 2 (due Mar 25), 3 (due Apr 17), 4 (due May 28)
Term paper guidelines and possible topics: [PDF]
Facts about the étale fundamental group: [PDF]
Feb 26 | Overview. The fundamental group of a complex variety (I) |
Mar 5 | The fundamental group of a complex variety (II) |
Mar 12 | The fundamental group of a complex variety (III) (guest lecture by Adrian Langer). |
Mar 19 | The étale fundamental group (I) |
Mar 26 | The étale fundamental group (II) |
Apr 2 | Faithfully flat descent |
Apr 9 | |
Apr 16 | Galois categories |
Apr 23 | The étale fundamental group (I) |
Apr 30 | The étale site |
May 7 | No lecture |
May 14 | The étale fundamental group (II) |
May 21 | The étale fundamental group (III) |
May 28 | The pro-étale fundamental group (I) |
Jun 4 | The pro-étale fundamental group (II) |
Jun 11 | Anabelian geometry (guest lecture by Sylvain Gaulhiac) |
Jun 18 | Term paper presentations + cake |
Contact: pachinger@impan.pl