Arithmetic Triple Linking Numbers (Slides)
Abstract: Following the analogy between knots and primes introduced by Mazur in the 1960s, we define multiple linking numbers of primes in a number field K. I will outline the proof of a result by Amano et al. which relates mod 2 triple linking numbers (also sometimes called the Rédei symbol) to L-functions and modular forms, which gives an explicit and constructive example of the theorem by Weil-Langlands and Deligne-Serre. To conclude I will discuss some recent work-in-progress and difficulties encountered in trying to generalise this to mod 3 triple linking numbers.
|Y-RANT V||8 September 2023|
What prime is your shoelace? The answer may surprise you! (Linking, Knots and Primes) (Handwritten Notes)
Abstract: In the 1960s Barry Mazur pointed out an analogy between the behaviour of prime ideals of a number field and knots in a 3-manifold. This observation birthed the field of Arithmetic Topology, which is the study of Number Theory through this perspective. The goal of this talk is to show the audience snippets of this analogy, in particular the analogy between linking numbers of knots and the power residue symbol of primes.
|Hodge Club Seminar||3 March 2023|
|An introduction to the étale fundamental group (Slides)||GLaMS Examples Seminar||9 December 2022|
Reading Groups and Pre-talks
|Separated and proper morphisms||Eisenboaties||26 May 2023|
Étale fundamental groups and étale cohomology
(Ahead of Eric Ahlqvist: Massey Products and Class Field Towers)
|Glasgow Algebra Pre-Seminar (GAPS)||24 May 2023|
The analogy between Knots and Primes, and a very brief introduction to Topological Quantum Field Theories (Handwritten Notes)
(Ahead of Minhyong Kim: Path Integrals and p-adic L-functions)
|Glasgow Algebra Pre-Seminar (GAPS)||1 March 2023|
|Dimension Theory||Eisenbuddies||17 February 2023|