Primes, knots and periodic orbits

TL;DR

This paper explores deep analogies between number theory, knot theory, and dynamical systems, and reports on constructing dynamical systems for arithmetic schemes to realize these analogies.

Contribution

It introduces a novel framework linking algebraic number theory and knot theory through dynamical systems for arithmetic schemes.

Findings

Established analogies between algebraic number theory and knot theory

Linked analytic number theory with dynamical systems

Constructed dynamical systems for arithmetic schemes

Abstract

This is a text written for the Ennio De Giorgi Colloquio volume. It covers analogies between algebraic number theory and knot theory, analogies between analytic number theory and certain dynamical systems, and a report on our construction of dynamical systems for arithmetic schemes which realize some of these analogies.