Abstract: Mazur first observed in the 60s a deep analogy between the embedding of a knot in a 3-manifold and primes in a number field. Witten showed that knot invariants can be obtained by computations from quantum field theory. Using ideas from this analogy, Minhyong Kim and his collaborators developed the study of arithmetic field theories. This talk will be an introduction to Arithmetic Field Theories, in particular focusing on Arithmetic Chern-Simons Theory.

Abstract: Mazur first observed in the 60s a deep analogy between the embedding of a knot in a 3-manifold and primes in a number field. Using ideas from this analogy, Minhyong Kim and his collaborators developed the study of arithmetic field theories. This talk will be an introduction to Arithmetic Field Theories, in particular focusing on Arithmetic Chern-Simons Theory.

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.

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.