Beijing Institute of Mathematical Sciences and Applications

Non-archimedean geometry provides a powerful framework for studying spaces defined over fields equipped with non-archimedean valuations, such as the field of p-adic numbers or the Novikov field. Unlike classical real or complex geometry, where the absolute value satisfies the usual triangle inequality, the non-archimedean setting is governed by the ultrametric inequality, leading to different analytic and geometric behavior.

This course offers a first introduction to the subject, beginning with the basic concepts of valued fields, non-archimedean norms, and analytic functions in this context. We then explore the construction and geometry of Berkovich analytic spaces. If time permits, we may discuss connections to mirror symmetry and especially to Strominger-Yau-Zaslow conjecture.

The goal is to provide students with both the foundational language and the intuition necessary to engage with current research directions. No prior background in non-archimedean geometry will be assumed, though familiarity with algebraic geometry or complex geometry will be helpful.