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.
Projectdetails
Introduction
R. Langlands conjectured the existence of a correspondence between automorphic spectrums of Hecke algebras and representations of Galois groups of global fields. The existence of such correspondence is one of the main conjectures in mathematics. Even if not known in full generality, it leads to proofs of Fermat and Sato-Tate conjectures.
Project Overview
This project is on three aspects of the Langlands correspondence.
First Aspect
The first part of this project is a description of the spectrum of Hecke algebras on the space generated by pseudo Eisenstein series of cuspidal automorphic forms of Levi subgroups. In the simplest non-trivial case, the precise description is a conjecture of Langlands.
This conjecture is proven in my work with A. Okounkov, by an unexpected topological interpretation. I expect this approach to work in a number of other cases.
Second Aspect
The second part of this project is an extension of the Langlands correspondence to a completely new area of fields of rational functions on curves over local fields.
This extension of the Langlands correspondence to a new area could lead to new interplays between Representation Theory and Number Theory.
Third Aspect
The third part of the project is on a categorification of the Langlands correspondence necessary for establishing the strong form of this correspondence.
Financiële details & Tijdlijn
Financiële details
Subsidiebedrag | € 1.976.875 |
Totale projectbegroting | € 1.976.875 |
Tijdlijn
Startdatum | 1-5-2024 |
Einddatum | 30-4-2029 |
Subsidiejaar | 2024 |
Partners & Locaties
Projectpartners
- THE HEBREW UNIVERSITY OF JERUSALEMpenvoerder
Land(en)
Vergelijkbare projecten binnen European Research Council
Project | Regeling | Bedrag | Jaar | Actie |
---|---|---|---|---|
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 |
Relative Langlands Functoriality, Trace Formulas and Harmonic AnalysisThe 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. | ERC Consolid... | € 1.409.559 | 2022 | Details |
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 |
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 |
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.
Relative 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.
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.
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.