Descripción del proyecto
Del programa general de Langlands al programa relativo Langlands, con nuevos ejemplos relevantes
El programa de Langlands, propuesto hace más de cincuenta años, se suele considerar como el proyecto más grande de las matemáticas modernas. Su autor, Robert Langlands, obtuvo el premio Abel 2018, uno de los premios más prestigiosos de las matemáticas. El programa de Langlands es una gran teoría unificada de las matemáticas que describe interacciones entre el álgebra, la teoría de números y el análisis armónico. La funtorialidad es un concepto básico que describe relaciones profundas entre los espectros locales y automórficos de diferentes grupos. El programa relativo de Langlands es una generalización reciente y prometedora del programa general de Langlands. El objetivo del proyecto RELANTRA, financiado con fondos europeos, es realizar contribuciones fundamentales al desarrollo de estos conceptos con un enfoque en el entorno local, lo que da lugar a nuevos ejemplos relevantes de correspondencias-funtorialidades del programa relativo de Langlands.
Objetivo
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.
Ámbito científico
Palabras clave
Programa(s)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Régimen de financiación
HORIZON-ERC - HORIZON ERC GrantsInstitución de acogida
75794 Paris
Francia