Title: Introduction to Berkovich analytic spaces
Speaker:Jérôme Poineau - Université de Caen
Date: Nov. 9, 10, 12, 13 and 16,  10am-12pm.  
Classroom: 82J12（11/12）、82J04（Other time）
Abstract: At the end of the eighties, Vladimir Berkovich introduced a new way to define p-adic analytic spaces. A surprising feature is that, although p-adic fields are totally discontinuous, the resulting spaces enjoy many nice topological properties: local compactness, local path-connectedness, etc. On the whole, those spaces are very similar to complex analytic spaces. They already have found numerous applications in several domains: arithmetic geometry, dynamics, motivic integration, etc.
In this course, we will give an introduction to p-adic geometry in the sense of Berkovich (although we will also mention other theories in passing). We will cover the following topics:
- Tate algebras and affinoid algebras
- affinoid spaces
- rigid spaces (Tate's theory)
- analytification of algebraic varieties (and thorough description of the affine line in the Berkovich setting)
- formal schemes and Raynaud's theory
- analytic curves
- étale cohomology and vanishing cycles