### Seminar in Algebra, Lie Theory, and Geometry

# Bayani Hazemach

### (Jacobs Unversity)

## "Galois groups and fundamental groups."

** Date: ** |
Thu, April 19, 2018 |

** Time: ** |
13:30 |

** Place: ** |
Seminar Room (120), Research I |

**Abstract:** (this talk follows the text of the same name by Szamuely) The main theorem of Galois theory will be reformulated in language introduced by Grothendieck, as a categorical equivalence between finite etale algebras over a given field and finite sets equipped with a continuous action by the absolute Galois group. An analogous topological development is given for covering theory, with the fundamental group playing the role of the absolute Galois group. Finally, we specialize to the case of Riemann surfaces, where these analogous descriptions are actually equivalent, so that for a given compact Riemann surface, we are provided a dictionary between finite Galois branched covers and finite etale algebras over the meromorphic function field. (This may actually be generalized to algebraic curves over an arbitrary perfect base field, which we will state with a bit more details if time permits.)