Beijing Institute of Mathematical Sciences and Applications

Berkovich spaces provide a powerful framework for doing analytic geometry over non-Archimedean fields, bridging the gap between rigid analytic geometry and more topological approaches. Introduced by Vladimir Berkovich in the 1990s, these spaces offer rich topological structures—locally compact, path-connected, and Hausdorff—that allow for intuitive interpretations of non-Archimedean phenomena.

In this introductory course, we will explore the motivations behind Berkovich's construction, starting from classical problems in p-adic geometry and the limitations of Tate’s rigid spaces. We will explain how Berkovich’s approach resolves issues of topology and local connectivity, and why it has become essential in modern arithmetic geometry, non-Archimedean dynamics, and the study of degenerations of algebraic varieties.