Galois Groups and Fundamental Groups

There are many formal analogies between the classical Galois theory of fields and the theory of covering spaces in algebraic topology. In this seminar we will see that, in certain situations over the complex numbers, this intuition can be turned into a theorem, using Riemann surfaces as a bridge between the two worlds. As an application of this theory, we will compute the absolute Galois group of the complex function field C(t). We will roughly follow the first three chapters of Szamuely's book "Galois Groups and Fundamental Groups" . No previous knowledge of covering spaces or Riemann surfaces will be required, as we will introduce these notions during the seminar. If you are already familiar with these, it will still be interesting for you to see how these notions interact from a Galois-theoretic viewpoint. We will have an organizatorial meeting on February 10. The zoom link is here. A detailed seminar program can be found here . Distribution of the Talks:- Infinite Galois Theorey (Alexander Schneider, April 19, notes of the talk)
- Interlude on Category Theory (Florian Greindl, April 26)
- Finite Étale Algebras (Alexander Feeß, May 3, notes of the talk)
- Basics of Covering Theory (Krishna Kumar Madhavan-Vijayalakshmi, May 10, notes of the talk)
- Galois Covers (Simon Lang, May 17, notes of the talk)
- Monodromy (Caspar Heusinger, May 31, notes of the talk)
- The Universal Cover (Benedekit Loehlein, June 7)
- Locally Constant Sheaves (Benni Ngo, June 14)
- Riemann Surfaces (Robert Stolz, June 21, notes of the talk)
- Relation with Field Theory (Changhao Wang, June 28)
- The Absolute Galois Groups of the Complex Function Field (Arshay Sheth, July 5)