SubsidieMeestersRelative Langlands Functoriality, Trace Formulas and Harmonic Analysis
The project aims to advance the relative Langlands program by developing local trace formulas and spectral expansions to establish new correspondences in number theory and representation theory.
Projectdetails
Introduction
The Langlands program is a web of vast and far-reaching conjectures connecting seemingly distinct areas of mathematics: number theory and representation theory. At the heart of this program lies an important principle called functoriality, which postulates the existence of deep relations between the automorphic representations of different groups, as well as related central analytic objects called automorphic L-functions.
The Relative Langlands Program
Recently, a new and particularly promising way to look at these notions, known as the relative Langlands program, has emerged. This approach essentially consists of replacing groups with certain homogeneous spaces and studying their automorphic or local spectra.
Importance of Trace Formulas
As with the usual Langlands program, trace formulas are essential tools in the relative setting. They are used both to tackle new conjectures and to deepen our understanding of the underlying principles.
Main Contributions
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:
- Studying systematically the spectral expansions of certain simple versions, especially in the presence of an outer automorphism (twisted trace formula).
- Developing far-reaching local relative trace formulas for general spherical varieties, making original new connections to the geometry of cotangent bundles.
These advancements would then be applied to establish new and important instances of relative Langlands correspondences/functorialities.
Additional Directions
In a slightly different but related direction, I also aim to study and develop other important tools of harmonic analysis in a relative context. This includes:
- Plancherel formulas
- New kinds of Paley-Wiener theorems
These tools may have possible applications to new global comparisons of trace formulas and the factorization of automorphic periods.
Financiële details & Tijdlijn
Financiële details
| Subsidiebedrag | € 1.409.559 |
| Totale projectbegroting | € 1.409.559 |
Tijdlijn
| Startdatum | 1-9-2022 |
| Einddatum | 31-8-2027 |
| Subsidiejaar | 2022 |
Partners & Locaties
Projectpartners
- CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRSpenvoerder
Land(en)
Vergelijkbare projecten binnen European Research Council
| Project | Regeling | Bedrag | Jaar | Actie |
|---|---|---|---|---|
Motives and the Langlands programThis project aims to develop new tools using motivic sheaves to address the Langlands program for function fields, enhancing the understanding of the relationship between automorphic forms and motives. | ERC Starting... | € 1.409.163 | 2022 | Details |
The Langlands CorrespondenceThis project explores the Langlands correspondence through Hecke algebras, extends it to rational functions on curves, and seeks a categorification to strengthen its foundational aspects. | ERC Advanced... | € 1.976.875 | 2024 | Details |
Modularity and Reciprocity: a Robust ApproachThis project aims to enhance understanding of Galois representations in the Langlands programme through innovative techniques, addressing key conjectures in arithmetic geometry and automorphic forms. | ERC Consolid... | € 1.473.013 | 2025 | Details |
Automorphic Forms and ArithmeticThis project seeks to advance number theory and automorphic forms by addressing three longstanding conjectures through an interdisciplinary approach combining analytic methods and automorphic machinery. | ERC Advanced... | € 1.956.665 | 2023 | Details |
Correspondences in enumerative geometry: Hilbert schemes, K3 surfaces and modular formsThe project aims to establish new correspondences in enumerative geometry linking Gromov-Witten and Donaldson-Thomas theories, enhancing understanding of K3 surfaces and modular forms. | ERC Starting... | € 1.429.135 | 2023 | Details |
Motives and the Langlands program
This project aims to develop new tools using motivic sheaves to address the Langlands program for function fields, enhancing the understanding of the relationship between automorphic forms and motives.
The Langlands Correspondence
This project explores the Langlands correspondence through Hecke algebras, extends it to rational functions on curves, and seeks a categorification to strengthen its foundational aspects.
Modularity and Reciprocity: a Robust Approach
This project aims to enhance understanding of Galois representations in the Langlands programme through innovative techniques, addressing key conjectures in arithmetic geometry and automorphic forms.
Automorphic Forms and Arithmetic
This project seeks to advance number theory and automorphic forms by addressing three longstanding conjectures through an interdisciplinary approach combining analytic methods and automorphic machinery.
Correspondences in enumerative geometry: Hilbert schemes, K3 surfaces and modular forms
The project aims to establish new correspondences in enumerative geometry linking Gromov-Witten and Donaldson-Thomas theories, enhancing understanding of K3 surfaces and modular forms.