Coverings in Topology, Geometry and Algebra. Spring 2017

Master course, Radboud University Nijmegen

Ben Moonen and Arne Smeets, Radboud University Nijmegen. Email: B.Moonen at science.ru.nl and arnesmeets at gmail.com

The lectures are given on Tuesdays from 10:45 till 12:30. **NOTE: in order to facilitate video recording of the lectures, the course has been rescheduled. Starting February 7 the lectures will be in room HG00.616.** The exercise classes are on Tuesdays from 15:45 till 17:30 in room HG00.058.

Topology, Algebra (Groups, Rings and Fields), basic Complex function theory. Familiarity with Galois theory is *not* assumed (but you may profit from it if you have seen Galois theory before). It is helpful if you have already seen some basic notions of category theory, though also this will be reviewed. It is *strongly recommended* that, in parallel with this course, you take the mastermath course on Riemann surfaces (if you have not already taken a course on Riemann surfaces or algebraic curves.)

The central theme in this course is the striking analogy between the notion of a covering in Topology and the notion of a field extension in Algebra. Thus, the slogan is: "Fundamental groups = Galois groups". This leads to surprising connections between two mathematical topics that at first glance may not appear to be so directly related.

In the course, we will discuss in detail the fundamental group, both in terms of homotopy classes of loops and in terms of topological coverings. Next we will discuss the main results in Galois theory, aiming at students for whom this is new material. After discussing a unified approach between these theories, we will apply these insights to Geometry, notably to the study of algebraic curves. As a nice application, we will discuss some results about the inverse Galois problem.

T. Szamuely, Galois groups and Fundamental Groups, Cambridge University Press.

We will sometimes also use some parts of Hatcher's book on Algebraic Topology.

Depending on the students's background, this may be supplemented by parts of other texts.

In the lectures we can only discuss part of the material in detail. Students in this course are supposed to study the book by Szamuely (or supplementary literature) by themselves. Each week some exercises will be suggested, and there is an exercise class on Tuesdays from 15:45 till 17:30, led by Salvatore Floccari.

During the semester, three hand-in assignments will be given. In order to pass, you need to obtain a sufficient score (6 or higher) for these assignments. The course is then concluded by an oral exam. For these oral exam, the following time slots are available, which are assigned on a first come, first served basis. You are requested to send us an email with your preferred day and time with one or two alternatives. If you are unable to take the oral exam on 26 June, let us know.

Day and time | Student |
---|---|

June 26, 12:00-12:45 | Sterre den Breeijen |

June 26, 12:45-13:30 | Lisa Noorlander |

June 26, 13:30-14:15 | Luuk Verhoeven |

June 26, 14:45-15:30 | Rosa Schwarz |

June 26, 15:30-16:15 | Robert Christian Subroto |

June 26, 16:15-17:00 | Maarten Smit |

Date | Topics | Exercises |
---|---|---|

January 31 | Section 1.1; Section 1.2 up until Lemma 1.2.2 | |

February 7 | Main theorem of Galois theory | |

February 14 | The Main Theorem for infinite Galois extensions | |

February 21 | Grothendieck's formulation of Galois theory | |

February 28 | no class | |

March 7 | The fundamental group | |

March 14 | Topological coverings | |

March 21 | Galois theory of topological coverings | |

March 28 | no class | |

April 4 | no class | |

April 11 | The Seifert-van Kampen theorem | Here is the 2nd homework assignment |

April 18 | Riemann surfaces | Vul svp even het volgende formulier in en stuur ons dat. (Je mag ook gewoon je antwoorden als tekst per email sturen.) |

April 25 | no class | |

May 2 | The topology of Riemann surfaces | |

May 9 | From algebraic curves to Riemann surfaces | |

May 16 | Riemann-Roch; projective embeddings of Riemann surfaces | |

May 23 | Fundamental groups and finite covers of Riemann surfaces | |

May 30 | Algebraic curves and function fields | Here is the 3rd homework assignment |

June 6 | Last lecture |

To the webpage of Ben Moonen.