The Laplacian on metrized graphs II
Let Γ be a metrized graph. Last semester, we explained how to construct a Laplacian operator Δ such that Δf is a finite signed Borel measure on Γ with total mass 0 whenever f is a real-valued function of Bounded Differential Variation on Γ.
The first goal of this course is to show the converse: if μ is a finite signed Borel measure on Γ with total mass 0, then μ is the Laplacian of some real-valued function with bounded differential variation on Γ.
Once this fact established, we will apply the theory of potential kernels to the Berkovich projective line, which is homeomorphic to a inverse of metrized graphs, and we will make connections with the Hsia kernel.
The first goal of this course is to show the converse: if μ is a finite signed Borel measure on Γ with total mass 0, then μ is the Laplacian of some real-valued function with bounded differential variation on Γ.
Once this fact established, we will apply the theory of potential kernels to the Berkovich projective line, which is homeomorphic to a inverse of metrized graphs, and we will make connections with the Hsia kernel.
Lecturer
Date
13th October ~ 29th December, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday | 13:30 - 16:55 | A3-2-301 | ZOOM A | 388 528 9728 | BIMSA |
Prerequisite
Even though I will review all the necessary prerequisites during the first class, it is better to have taken my course 'The Laplacian on metrized graphs'.
Syllabus
Class 1: Remainders from my last course
Class 2 - Class 6: we will show that if μ is a finite signed Borel measure on Γ with total mass 0, then μ is the Laplacian of some real-valued function with bounded differential variation on Γ.
Class 7 : Remainders from the Berkovich projective line
Class 8- Class 12: Study of the Hsia Kernel on the Berkovich projective line and connection with the potential kernels.
Class 2 - Class 6: we will show that if μ is a finite signed Borel measure on Γ with total mass 0, then μ is the Laplacian of some real-valued function with bounded differential variation on Γ.
Class 7 : Remainders from the Berkovich projective line
Class 8- Class 12: Study of the Hsia Kernel on the Berkovich projective line and connection with the potential kernels.
Reference
The main reference will be Chapter 3 (Subsections 3.6 and 3.7) and Chapter 4 (Subsections 4.1 and 4.2) of Baker and Rumely's book "Potential Theory and Dynamics on the Berkovich projective line".
Audience
Undergraduate
, 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.