Le Séminaire différentiel se tiendra au LMV toute la journée.  Cliquer ici pour plus d’informations.

Programme de la journée

10:00 Accueil des participants

10:30-11:30 — Jean-Pierre Ramis
A new spectral theory: A complex analytic intrinsic theory with applications to the Connes-Moscovici spectrum, which matches the zeros of zeta and to linear perturbations of black-holes
I propose a new paradigm for the spectral theory of ODEs with analytic coefficients.
The classical approach is based on Hilbert spaces and linear operators (non necessarily bounded). My approach is based on analytic continuation, in the complex domain, and

-summability, otherwise speaking on wild monodromy.
My analytic spectra are intrinsic, the boundary conditions are « contained in the equation ». I recently discovered that this last concept has been clearly formulated by Erwin Schr\ »odinger in a letter of december 27-th in relation with the hydrogen spectrum.
The analytic spectra are compatible with some « natural transformations » of ODEs as -homotopic transformations and gauge transformations. There are strong relations with special functions theory and the spectra appearing in natural sciences are in a lot of cases analytic spectra. There are numerous examples in quantum chemistry and in black holes theory.
I will explain my understanding of the ideas of Schr\ »odinger on the hydrogen spectrum at the end of 1925. It is a first version of my notion of spectrum. After, I will limit myself to the analytic spectra of the Confluent Heun Equations (CHE). The CHE are the rational linear second order ODEs admitting two regular singular points and an irregular singular point.
The analytic spectra are defined by the fact that the eigenfunctions are special solutions connecting local special solutions a two singular points. These local special solutions are, by definition,  eigenfunctions of the monodromy at a regular singular points and sum of an eigenfunction of the exponential torus (formal pure wave) at an irregular singular points.
I will give some results on the generalised polynomial solutions of the CHE. Such solutions are defined by the fact that their logarithmic derivative is rational. Equivalently they connect special local solutions. One get them using the first step () of the Kovacic algorithm.
I will present a new surprising result about the existence of algebraic one parameter families of CHEs admitting a basis of generalised polynomial solutions.
I will give a new definition of the Connes-Moscovici spectrum, which matches the zeros of zeta, as an analytic bi-irregular spectrum and some consequences.
I will explain how analytic spectra appear in the theory of perturbations of black-holes, in relation with the Quasi-Normal-Modes (QNM) and with the algebraically special solutions. In this last case our results on the CHE admitting a basis of generalised polynomial solutions give a new light on the underlying physics.
Joint works with Anne Duval, Michè\`ele Loday-Richaud, Emmanuel Paul, Françoise Richard-Jung and Jean-Thomann.

[Déjeuner]

14:15-15:15 — Jacques Sauloy,
Classification analytique des équations aux q-différences et fibrés vectoriels holomorphes.
Soit un complexe tel que . À tout module aux -différences analytique est associé un fibré vectoriel holomorphe sur la courbe elliptique associée (courbe de Tate). Pour les moduli fuchsiani, ceci permet la classification analytique et même le calcul du groupe de Galois (Baranovsky-Ginzburg-Kontsevich). Pour les modules aux -différences analytiques quelconques, on obtient la classification formelle (van der Put-Reversat-Atiyah). Pour avoir la classification analytique de modules quelconques (équations irrégulières), il est nécessaire (et suffisant) de prendre en compte la filtration canonique par les pentes du côté des modules ; et une notion de « filtration anti-HN » (HN pour Harder-Narasimhan) du côté des fibrés : c’est une variante d’un énoncé attribué à Ramis-Sauloy-Zhang par Kontsevich-Soibelman.
If (and only if) necessary and appropriate, the talk will be given in English.

15:15-16:15 — Jacques-Arthur Weil
Reduced Forms and Galois Groups of Differential or Functional Systems
Consider a square matrix with entries in a differential field and the linear differential system . This system is said to be in reduced form when is in , where

is the Lie algebra of the differential Galois group. Putting a system into reduced form is a technique that has been developed over the last fifteen years to determine the Galois group (or information about transcendence) by several authors, including the speaker. In this presentation, I will explain the main techniques for determining a reduced form, some recent progress, and some ongoing questions.