Introduction to p-adic differential equations
In the ultrametric setting, linear differential equations exhibit phenomena not observed in the complex field. Indeed, the solutions to these equations may not converge everywhere, even in the absence of poles. This introduces a non-trivial concept of the radius of convergence, and understanding it allows us to gain several interesting insights about the equation. Notably, it influences the finite dimensionality of de Rham cohomology. The main goal of this course is to introduce the framework of ultrametric differential equations.
Lecturer
Date
21st March ~ 21st June, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Thursday,Friday | 15:20 - 16:55 | A3-2a-302 | ZOOM 04 | 482 240 1589 | BIMSA |
Prerequisite
General algebra, Linear algebra, Galois theory of fields, Metric spaces.
Syllabus
The cours can be divided onto the main following parts:
- Some functional ultrametric analysis;
- Extensions of an ultrametric field;
- Generality on differential modules;
- Radii of convergence of the solutions;
- Decomposition with respect to the radii of convergence of the solutions.
- Some functional ultrametric analysis;
- Extensions of an ultrametric field;
- Generality on differential modules;
- Radii of convergence of the solutions;
- Decomposition with respect to the radii of convergence of the solutions.
Reference
S. Bosch, U. Güntzer, and R. Remmert. Non-archimedean analysis : a systematic approach to rigid analytic geometry. Grundlehren der mathematischen Wissenschaften, 261. Springer-Verlag, 1984.
K. S. Kedlaya. p-adic differential equations. English. Cambridge: Cam- bridge University Press, 2010.
G. Christol. “Le théorème de turritin p-adique (version du 11/06/2011)”.
G. Christol. “Modules différentiels et équations différentielles p-adiques”. In: Queen’s Papers in Pure and Applied Math (1983).
G. Christol and P. Robba. Équations différentielles p-adiques - Applications aux sommes exponentielles. Actualités Mathématiques. Hermann, 1994.
K. S. Kedlaya. p-adic differential equations. English. Cambridge: Cam- bridge University Press, 2010.
G. Christol. “Le théorème de turritin p-adique (version du 11/06/2011)”.
G. Christol. “Modules différentiels et équations différentielles p-adiques”. In: Queen’s Papers in Pure and Applied Math (1983).
G. Christol and P. Robba. Équations différentielles p-adiques - Applications aux sommes exponentielles. Actualités Mathématiques. Hermann, 1994.
Audience
Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
I have been an assistant professor at BIMSA since January 2024. My research primarily focuses on p-adic differential equations. I defended my Ph.D. thesis in 2018 at Montpellier University. Before joining BIMSA, I was an assistant professor at Algiers University. Subsequently, I was a postdoc at YMSC, Tsinghua University, from April 2021 to December 2023.