# Seminar: The Arithmetics of the Hyperbolic Plane

Sommersemester 2017

## Daniele Alessandrini

## Beatrice Pozzetti

### Schedule

- Tuesday 16:00-18:00 Uhr, INF 205, Room SR 4

### Müsli

Please, register for the seminar on MÜSLI, and send us an e-mail (daniele.alessandrini@gmail.com and pozzetti@mathi.uni-heidelberg.de).

### Description

The seminar will focus on the interplay between elementary number theory and hyperbolic geometry. We will study the Farey tassellation of the hyperbolic plane, and discuss how it can be used to understand number theoretic concepts like pythagorean triples, continued fractions, Euclidean algorithm, circle packings. In the other direction we will see how quadratic extensions of Q can be used to construct hyperbolic surfaces and how continued fractions give insights on the hyperbolic geodesic flow.

At the organizational meeting several topics will be proposed. Among them, you can choose one topic you are interested in. Once you have agreed on the topic of your project with us, you should contact us to discuss the content of your talk in more detail. Please let us help you during the preparation of your presentation. Every student is expected to hand in a short report as well.

Here is a syllabus of the talks.

### Prerequisites

This seminar is aimed at students who are interested in differential geometry. Students are expected to have a good knowledge of calculus and a certain familiarity with the differential geometry of manifolds. There are several topics for students to choose from, and hence any interested students are welcome to attend. The seminar is open for bachelor and master students. The seminar will be taught in English.

### Schedule of the talks

04.04.2017 | organization meeting | |

18.04.2017 | organization meeting | |

02.05.2017 | Valentino | Hyperbolic geometry |

09.05.2017 | Oskar | The Farey diagram |

16.05.2017 | Clemens | Continued franctions and cutting sequences-animations |

06.06.2017 | Daniele | Existence of dense geodesics |

20.06.2017 | Ferdinand | Quadratic forms |

27.06.2017 | Manuel | Classification of Quadratic forms |

11.07.2017 | Beatrice | Ford circle packings |

18.07.2017 | Menelaos | Quadratic number fields and discrete subgroups of PSL(2, R) |

### References

- Hatcher: Topology of numbers
- Witte-Morris: Introduction to arithmetic groups
- Bonahon: Low dimensional geometry

Zuletzt geändert: 29/08/2017