Project description DEENESFRITPL Laying the foundations for the representation theory of p-adic groups Representation theory studies abstract algebraic structures by representing their elements as linear transformations of vector spaces and investigates how they act on vector spaces. A fundamental problem of this theory is the construction of all representation types (irreducible, smooth, complex) of certain matrix groups, called p-adic groups. Despite much progress in the field over the last 40 years, surprisingly little is known about these representations in the general setting. The p-adic number systems play a fundamental role in number theory and in other parts of mathematics, offering a wide number range of settings to explore questions about rational numbers. The EU-funded GReatLaP project aims to construct all representations in full generality. The project will then extend the representation theory of p-adic groups to the study of the global and relative Langlands programme. Show the project objective Hide the project objective Objective My objectives consist of laying new foundations for the representation theory of p-adic groups and making significant progress on the local, global and relative Langlands program.The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. Work in this area has also lead to the resolution of other major conjectures including Fermat's Last Theorem.A fundamental problem on the representation theory side is the construction of all (irreducible, smooth, complex) representations of certain matrix groups, called p-adic groups. Despite much progress in the past 40 years, we still know surprisingly little about these representations in the general setting. My first main objective is the construction of all (supercuspidal) representations in full generality. This will form the foundation for the future of the representation theory of p-adic groups and have a plethora of applications also beyond this area. Solving this problem will involve tackling all the complications that arise in the non-tame case compared to the tame case.I will then demonstrate the power of this result beyond the representation theory of p-adic groups by making significant contributions to the - global Langlands program. This will be achieved by constructing congruences between automorphic forms based on the existence of enough suitable (omni-)supercuspidal types for p-adic groups.- relative Langlands program. I will prove finite multiplicity of the representations occurring in the space of function on a spherical variety by combining my results about the shape of representations with properties of the moment map.Finally, I will use my insights to advance the explicit local Langlands correspondence by proving that the most-general construction to date, which treats non-singular representations, satisfies all required properties and suggesting a correspondence beyond non-singular representations. Fields of science natural sciencesmathematicspure mathematicsarithmeticsnatural sciencesmathematicsapplied mathematicsnumerical analysis Programme(s) H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC) Main Programme Topic(s) ERC-2020-STG - ERC STARTING GRANTS Call for proposal ERC-2020-STG See other projects for this call Funding Scheme ERC-STG - Starting Grant Coordinator RHEINISCHE FRIEDRICH-WILHELMS-UNIVERSITAT BONN Net EU contribution € 1 499 491,00 Address Regina pacis weg 3 53113 Bonn Germany See on map Region Nordrhein-Westfalen Köln Bonn, Kreisfreie Stadt Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Other funding € 0,00 Beneficiaries (2) Sort alphabetically Sort by Net EU contribution Expand all Collapse all RHEINISCHE FRIEDRICH-WILHELMS-UNIVERSITAT BONN Germany Net EU contribution € 1 499 491,00 Address Regina pacis weg 3 53113 Bonn See on map Region Nordrhein-Westfalen Köln Bonn, Kreisfreie Stadt Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Other funding € 0,00 THE CHANCELLOR MASTERS AND SCHOLARS OF THE UNIVERSITY OF CAMBRIDGE Participation ended United Kingdom Net EU contribution € 0,00 Address Trinity lane the old schools CB2 1TN Cambridge See on map Region East of England East Anglia Cambridgeshire CC Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Other funding € 0,00
