Beijing Institute of Mathematical Sciences and Applications

Non-archimedean geometry studies spaces defined over fields equipped with a non-archimedean valuation, such as the field of p-adic numbers and the Novikov field. The ultrametric nature of these valuations gives rise to a form of analysis and geometry that differs markedly from the classical real and complex settings, while at the same time revealing deep connections with algebraic geometry, topology, and symplectic geometry.

This course will introduce the basic ideas and techniques of the subject. We will begin with valued fields, non-archimedean norms, and the foundations of non-archimedean analysis, including power series and analytic functions. We then move on to affinoid algebras and their associated analytic spaces, which provide the basic building blocks of the theory. From there, we develop the global geometric picture, with particular emphasis on Berkovich spaces and their topological and geometric properties, illustrated through examples.

If time permits, we will explore connections with tropical geometry and mirror symmetry, with special attention to the Strominger-Yau-Zaslow (SYZ) conjecture and related ideas in symplectic geometry.

The course is intended as a first introduction to the subject. No prior knowledge of non-archimedean geometry will be assumed, although some familiarity with algebraic or complex geometry will be very helpful.