Project description
Extending the scope of the Langlands correspondence
The Langlands correspondence, one of the main conjectures in mathematics, has been called a unified theory of mathematics. Funded by the European Research Council, the Correspondence project aims to investigate three aspects of this correspondence: the first is a general description of the spectrum of Hecke algebras on the space generated by pseudo Eisenstein series of cuspidal automorphic forms of Levi subgroups, something the investigator has proven in the simplest case. The second is an extension of the Langlands correspondence to a completely new area that could lead to new interplays between representation theory and number theory. The final aspect is a categorification of the Langlands correspondence needed to establish its strong form.
Objective
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 Ferma and Sato-Tate conjectures.
This project is on three aspects of the Langlands correspondence. 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.
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.
The third part of the project is on a categorification of the Langlands correspondence necessary for establishing the strong form of this correspondence.
Fields of science (EuroSciVoc)
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CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques.
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Keywords
- Langlands
- Representations
- Categorification
- G-equivariant bordism
- local fields
- automorphic forms
- 1
- 2- forms
- Hecke correspondence
- Hecke operators
- categorical trace
- symmetric infinity categories
- nilpotent singular support
- Eisenstein series
- the Springer stack
- Opers
- reductive group
- global and local fields
- dual group
- curves
- global differential operators on the stack of $G$-bundles
- cohomologically proper quotient stack
Programme(s)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Topic(s)
Funding Scheme
HORIZON-ERC - HORIZON ERC GrantsHost institution
91904 Jerusalem
Israel