Knot theory and Knot theory

In kindergarten, everybody learns Mazur’s analogy between number fields and 3-manifolds, where a number field is thought of as space of cohomological dimension 3 via Artin-Verdier duality and where primes in $\mathcal{O}_K$ correspond to knots in the resulting space. This analogy is currently only an analogy, there is no known way of associating a 3-manifold to a number field. There is however a more explicit relationship between knots and number theory in a different direction: in 2006, Etienne Ghys showed that there is a relationship between modular knots (related to integral binary quadratic forms) and closed periodic orbits (i.e. knots) of the Lorenz attractor. In this talk, we will explain the proof of this result.