Description du projet
Du programme de Langlands à son homologue relatif, avec de nouveaux cas importants
Le programme de Langlands, proposé il y a plus d’un demi-siècle, est souvent considéré comme le plus grand projet des mathématiques modernes. Récompensé par le prix Abel 2018, l’une des récompenses les plus prestigieuses dans le domaine des mathématiques, il s’agit d’une grande théorie mathématique unifiée décrivant des connexions étendues entre l’algèbre, la théorie des nombres et l’analyse. La functorialité est un concept central, qui décrit les relations profondes entre les spectres locaux et automorphes de différents groupes. Le programme de Langlands relatif est une généralisation récente et prometteuse du programme de Langlands classique. Le projet RELANTRA, financé par l’UE, entend apporter des contributions fondamentales au développement de ces concepts en mettant l’accent sur le cadre local, ce qui débouchera sur de nouveaux cas pertinents de correspondances/functorialités de Langlands relatives.
Objectif
The Langlands program is a web of vast and far-reaching conjectures connecting seemingly distinct areas of mathematics that are number theory and representation theory. At the heart of this program lies an important principle called functoriality, that postulates the existence of deep relations between the automorphic representations of different groups, as well as related central analytic objects called automorphic L-functions. Recently, a new and particularly promising way to look at these notions, and that has come to be called the relative Langlands program, has emerged. It essentially consists in replacing groups by certain homogeneous spaces and to study their automorphic or local spectra. As for the usual Langlands program, trace formulas are essential tools in the relative setting both to tackle new conjectures than to deepen our understanding of the underlying principles. A main theme of this proposal would be to make fundamental new contributions to the development of these central objects in the local setting notably by: (1) Studying systematically the spectral expansions of certain simple versions especially in the presence of an outer automorphism (twisted trace formula) (2) Developing far-reaching local relative trace formulas for general spherical varieties making in particular original new connections to the geometry of cotangent bundles. These progress would then be applied to establish new and important instances of relative Langlands correspondences/functorialities. In a slightly different but related direction, I also aim to study and develop other important tools of harmonic analysis in a relative context, including Plancherel formulas and new kind of Paley-Wiener theorems, with possible applications to new global comparison of trace formulas and factorization of automorphic periods.
Champ scientifique
Programme(s)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Régime de financement
HORIZON-ERC - HORIZON ERC GrantsInstitution d’accueil
75794 Paris
France