During the past 25 years there has been a significant activity constructing and analyzing C*-algebras and C*-algebraic dynamical systems from semigroups of number theoretic origin. Among these are the semigroup C*-algebras of the affine semigroup of the ring of algebraic integers in a number field, which are closely related to the Bost-Connes systems giving a thermodynamic interpretation of the Riemann and Dirichlet zeta-functions. The K-theory of the associated boundary quotients has been computed and also the phase transition of equilibrium states at low temperature has been exhibited via a natural parametrization. There is an intriguing, and still not understood, parallel in that the same C*-algebra lies at the heart of the solution of both problems. The goal of the master class is to introduce the participants to these and related topics lying at the intersection of several branches of mathematics.

The master class is a part of the project "Pure Mathematics in Norway, 2018-2022", supported by the Bergen Research Foundation and the TromsÃ¸ Research Foundation.