Analytic Manifolds and Berkovich Spaces

This lecture covers the theory of analytic manifolds, focusing on the few theorems from differential geometry that have analogues in this setting. It introduces Berkovich spaces, which are associated with rings over K-algebras, and explains the concept of multiplicative semi-norms. The lecture also discusses the Berkovich affine line AB as a set of multiplicative semi-norms extending the norm on K, along with the topology on the infinitely branched tree.