## a mini-course on Berkovich Spaces

**Speaker：** Morgan Brown (University of Michigan)

**Time：** 00:00-00:00 June 8, 2015 - June 11, 2015

**Venue：** Room 82J12, Building Jiayibing, Jingchunyuan 82, BICMR

Room: Room 82J12, Building Jiayibing, Jingchunyuan 82, BICMR

Time: June 8,10,11 2pm - 3pm.

Speaker: Morgan Brown (University of Michigan)

I: Introduction to Berkovich Spaces

A Berkovich space is a type of analytic space associated to an algebraic variety over a field $K$ with valuation $v$, such as $\mathbb{Q}_p$ or $\mathbb{C}((t))$. These spaces are intimately connected with other areas of mathematics, including tropical geometry, number theory, and more recently, birational geometry. I will give an introduction to the theory of Berkovich spaces, with an emphasis on two complementary ways of viewing these spaces: as spaces of valuations and as limits of finite simplicial complexes.

II: Skeleta and Dual Complexes

Berkovich spaces can be difficult to visualize. For example, the Berkovich space of $\mathbb{P}^1$ has the structure of an infinitely branching tree. It is often better to find a distinguished finite simplicial complex inside the Berkovich space, called a skeleton, which should reflect a great deal of the geometry of the ambient space. Usually these skeleta are given as dual complexes of various snc pairs and so can be studied using techniques from birational geometry.

III: Homotopy type of Berkovich spaces

I will present recent results on the homotopy type of some Berkovich spaces over $\mathbb{C}((t))$, based on the theorems and techniques presented in my first two lectures.=