Beijing Institute of Mathematical Sciences and Applications

Hansheng Diao , Yueke Hu , Emmanuel Lecouturier , Cezar Lupu

Tinhinane Azzouz

Tuesday, June 14, 2022 4:00 PM - 5:00 PM

1110

Zoom 361 038 6975 (BIMSA)

In the ultrametric setting, linear differential equations present phenomena that do not appear over the complex field. Indeed, the solutions of such equations may fail to converge everywhere, even without the presence of poles. This leads to a non-trivial notion of the radius of convergence, and its knowledge permits us to obtain several interesting information about the equation. Notably, it controls the finite dimensionality of the de Rham cohomology. In practice, the radius of convergence is really hard to compute and it represents one of the most complicated features in the theory of p-adic differential equations. The radius of convergence can be expressed as the spectral norm of a specific operator and a natural notion, that refines it, is the entire spectrum of that operator, in the sense of Berkovich. In our previous works, we introduce this invariant and compute the spectrum of differential equations over a power series field and in the $p$-adic case with constant coefficients. In this talk we will discuss our last results about the shape of this spectrum for any linear differential equation, the strong link between the spectrum and all the radii of convergence, notably a decomposition theorem provided by the spectrum.

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.