Oberseminar Algebra und Geometrie

Sommersemester 2019: Knots and Primes

Organised by I. Halupczok, B. Klopsch, S. Schröer and M. Zibrowius.

All talks take place Fridays at 12:30 in 25.22.03.73.

Short description

There exists an analogy by which prime numbers correspond to knots. The goal of the oberseminar is to understand this, by following the book "Knots and Primes" by Morishita.

A knot is an embedding $S^1 \to \mathbb{R}^3$. The analogy says that this corresponds to the embedding $\operatorname{spec} \mathbb{F}_p \to \operatorname{spec} \mathbb{Z}$. One of the first things we will see is that indeed, the fundamental groups on both sides correspond, where on the primes side, we work with étale fundamental groups. Another example of correspondence we shall see is that the Legendre symbol (of two primes) corresponds (more or less) to the linking number (of two knots).

Schedule

5.4.	1. Basic knot theory and the knot group (Benjamin Klopsch)
12.4.	2. Coverings and knots (Benno Kuckuck)
26.4.	3. Finite etale coverings (Johannes Fischer)
3.5.	4. The etale fundamental group (Leif Zimmermann)
10.5.	5. Number rings (Hamed Khalilian)
17.5.	6. The prime group (Benedikt Schilson)
24.5.	7. Introduction to class field theory (Matteo Vannacci)
31.5.	8. Local class field theory (Kevin Langlois)
7.6.	9. $\operatorname {spec}\mathbb {Z}$ is 3-dimensional (Thuong Dang)
14.6.	10. Guest talk: Vector fields and moduli of canonically polarized surfaces in positive characteristics (Nikolaos Tziolas; Cyprus)
21.6.	11. Overview over the analogy between primes and knots (Marcus Zibrowius)
28.6.	12. Decomposition of knots and primes (Stefan Schröer)
5.7.	13. Linking number and Legendre Symbols (Moritz Petschick)
12.7.	Program discussion for the next semester

→ detailed program

Literature

[Mor]	Morishita: Knots and Primes.
[Li]	Li: Knots and Primes.