Rigid analysis has a fruitful result in this decade. A not-so abstract example is the so-called Berkovich spaces or Berkovich trees. As a set, Berkovich space collects all seminorms on a function field over a non-archimedean field k. The topology on k is bad for analysis since it is disconnected. By creating the correspondent Berkovich space, we can embed k into the space. Berkovich space is connected and so is suitable for studying analytic problems. On the other hand, a perfectoid space is a whole new idea proposed and defined by P. Scholze. It provides a different topological advantage to build a bridge to connect some spaces over a characteristics 0 field and one over a characteristic p field. This connection allows us to pass many trivial results over characteristic p to characteristic 0.

 

In this talk, we will provide you with some graphs to help you imagine Berkovich spaces and perfectoid spaces, and we will talk about their application in arithmetic dynamics.