Il n’y aura pas d’exposé au séminaire AG.  Voir ici pour les informations sur le séminaire différentiel.

• 10:30-11:30: Florian Fürnsinn, Fuchs’ Theorem, an Exponential Function, and Abel’s Problem in Positive Characteristic
  Abstract: In the 19th century Fuchs and Frobenius developed a local solution theory for regular singular ordinary linear differential equations with complex polynomial coefficients. The Grothendieck -curvature conjecture motivates the search for a similar theory for equations whose coefficients are polynomials over fields of positive characteristic. In this talk I will define a differential extension of the power series over a field of positive characteristic , making use of new variables behaving under differentiation like iterated logarithms in characteristic zero. Every regular singular ordinary linear differential equation with polynomial or power series coefficients over admits a full basis of solutions in this extension. In particular, the exponential differential equation has a solution . Such solutions have remarkable properties, which we will explore. For example, I will discuss an analogue of Abel’s problem about the algebraicity of logarithmic integrals over. This talk is based on joint work with H. Hauser and H. Kawanoue.

• 12:00-14:00: repas.

• 14:15-15:15: Claudia Fevola, Euler discriminant of hyperplane arrangements

  Abstract: The Euler discriminant describes the locus of coefficients that cause a drop in the Euler characteristic of a very affine variety. In this talk, we focus on the case where the variety is the complement of hyperplanes. I will present formulas for two specific scenarios: when the coefficients are sparse and when they are restricted to a subspace of the parameter space. These formulas enable the computation of singularities in Euler integrals of linear forms, with applications in cosmology. This is joint work with Saiei Matsubara-Heo.

• 15:30-16:30: Antoine Chambert-Loir, Rationality and potential
  Abstract: A 1894 theorem by Émile Borel asserts that a power series with integral coefficients that defines a meromorphic function on a disk of radius > 1 is the Taylor expansion of a rational function. It has been extended in various directions (Pólya, Dwork, Bertrandias and Robinson) to encompass more complicated shapes than open disks, number fields, and several absolute values. We extend to algebraic curves of arbitrary genus the theorem of Cantor that considers Taylor expansions “at several points”. Our proof runs in two steps. The first step is an algebraicity criterion, which is proved using a method of diophantine approximation. The second step relies on the Hodge index theorem in Arakelov geometry, following an earlier work by Bost and myself. (Joint work with Camille Noûs) “Potentiel et rationalité”, arXiv:2305.17210