Descrizione del progetto
Dal programma Langlands al programma Langlands relativo con nuove, importanti istanze
Il programma Langlands, proposto oltre 50 anni fa, è spesso considerato uno dei più grandi progetti della moderna matematica. Insignito del premio Abel 2018, uno dei riconoscimenti più prestigiosi della matematica, si tratta di una grande teoria unificante di questo campo che descrive connessioni di ampia portata tra l’algebra, la teoria dei numeri e l’analisi. La funtorialità è un concetto centrale che traccia relazioni profonde tra gli spettri locali e automorfi di diversi gruppi. Il programma Langlands relativo rappresenta una promettente recente generalizzazione del programma Langlands classico. Il progetto RELANTRA, finanziato dall’UE, intende apportare contributi fondamentali allo sviluppo di tali concetti concentrando l’attenzione sulle configurazioni locali, conducendo a nuove istanze significative delle corrispondenze/funtorialità di Langlands.
Obiettivo
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.
Campo scientifico
Programma(i)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Argomento(i)
Meccanismo di finanziamento
HORIZON-ERC - HORIZON ERC GrantsIstituzione ospitante
75794 Paris
Francia