Scientific Scope
The EFI axis is structured around the theme of functional equations, studied from the perspectives of differential geometry, singularities, but also logic, combinatorics, number theory, and algorithmics. Due to this multidisciplinary nature, it is a cross-cutting axis spanning several RTs. Regarding the part of the EFI axis within the RT GAS, interactions with algebraic geometry and singularity theory mainly occur within the framework of the geometry of holomorphic foliations and moderate geometry (o-minimal, quasianalytical, etc.), where algebraic-geometrical considerations can be combined with information from either functional transcendence or asymptotic analysis (often, algebraic, differential, and functional equations are the basis for these interactions).
Many researchers in France work in fields where functional equations are either the subject of study or important tools for applications. These fields include “pure” and “applied” mathematics, computer science, and theoretical physics. In fact, there are currently increasingly strong interactions between theory and applications, with many joint actions having taken place over the past decade.
The scientific committee expects strong interactions between the EFI axis and the Singularities axis, particularly within the framework of o-minimal geometry. Indeed, in most cases, the construction of o-minimal structures uses a form of resolution of singularities adapted to quasianalytical classes, whereas the polynomially bounded case is well understood. The exponential case still needs to be explored from the perspective of singularities. In particular, there is an interest in addressing aspects of Hilbert’s 16th problem using o-minimal methods, and for this, it seems necessary to develop a quasianalytical geometry not only adapted to the asymptotic scale of real powers but also, more generally, to transasymptotic scales.
Website: https://efi.math.cnrs.fr/accueil.html
Genesis of the Functional Equations and Interactions Axis
The Functional Equations and Interactions axis is a continuation of the GDR EFI, created in 2019, which ended on December 31, 2023. More information can be found at this link, the former GDR website.