Abstract: Inspired by the Arithmetic Topology philosophy that primes in a number ring should be analogous to knots in a 3-manifold, Arithmetic Quantum Field Theories were first developed by Minhyong Kim as a way to generate arithmetic invariants.

In this talk I will introduce the basic ideas behind Arithmetic QFTs, and pairings of cohomology classes that arise from particular AQFTs. Time permitting, I will give examples of computations of these pairings.

Abstract: Mazur first observed in the 60s a deep analogy between knots in a 3-manifold and primes in a number field. In the 80s Witten showed that knot invariants can be computed using path integrals coming from quantum field theory. More recently, Minhyong Kim and his collaborators combined these ideas to develop the study of arithmetic field theories in order to compute new arithmetic invariants.

In this talk I will introduce an Arithmetic QFT known as Arithmetic Chern-Simons Theory, and define a notion of arithmetic linking numbers, analogous to the linking numbers of knots.

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.