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Qiuzhen College, Tsinghua University
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BIMSA > An introduction to the Berkovich Projective line and complex p-adic dynamics \(ICBS\)
An introduction to the Berkovich Projective line and complex p-adic dynamics
Let $p$ be any place of the rational field $\mathbb{Q}$ (can be the infinite place) and denote by $\mathbb{Q}_p$ the completion of $\mathbb{Q}$ with respect to the $p$-adic absolute value. We then set $C_p$ to be the completion of an algebraic closure of $\mathbb{Q}_p$. When $p$ is the infinite place of $\mathbb{Q}$, the field $C_p$, which is then the usual complex plane, is topologically nice (Haussdorf, simply connected, locally compact, spherically complete). However, when $p$ is a finite place, that is, a rational prime, the field $C_p$ is topologically "awful" (it is Haussdorf and complete, but totally disconnected, not locally compact and not spherically complete).

In the first part of this course, we will see that the Berkovich affine line, defined as the set of semi-norms on the ring of polynomials $C_p[X]$, turns $C_p$ into a "nice" topological space. More precisely, we will show that it is Haussdorf, locally compact, uniquely path-connected (so, it's a tree) and contains $C_p$ as a dense subset. After studying the topological properties of this space, we will start the study of complex $p$-adic dynamics on the Berkovich projective line, which is the "perfect" non-archimedean analogue of the classical complex dynamics.
Professor Lars Aake Andersson
Lecturer
Arnaud Plessis
Date
1st April ~ 17th June, 2025
Location
Weekday Time Venue Online ID Password
Tuesday 13:30 - 16:55 A3-2-201 ZOOM 02 518 868 7656 BIMSA
Prerequisite
none
Syllabus
Class 1-7 : Construction and topological study of the projective Berkovich line.

Class 8-12 : Dynamics on the projective Berkovich Line
Reference
The main reference of this course is Baker and Rumely's book entitled "Potential Theory and Dynamics on the Berkovich Projective Line".
Audience
Advanced Undergraduate , Graduate , Postdoc , Researcher
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Arnaud Plessis is an assistant professor at BIMSA from September 2023. His research is mainly focused on diophantine geometry. He obtained his Phd. thesis in 2019 at Université de Caen Normandie. Before joining BIMSA, he has been Attaché Temporaire d'Enseignement et de Recherche (a kind of postdoctoral with course duties) at Université Grenoble Alpes from September 2019 to August 2020. Then, he has been postdoctor at Morningside Center of Mathematics, Chinese Academy of Sciences, from September 2020 to August 2023.
Beijing Institute of Mathematical Sciences and Applications
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