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Details of Grant 

EPSRC Reference: EP/V001930/1
Title: Integral Structures in the Langlands Programme
Principal Investigator: Kurinczuk, Dr R
Other Investigators:
Researcher Co-Investigators:
Project Partners:
EC via Imperial College London Queen Mary University of London University of Vienna
Department: Pure Mathematics
Organisation: University of Sheffield
Scheme: EPSRC Fellowship
Starts: 01 June 2021 Ends: 31 May 2026 Value (£): 739,381
EPSRC Research Topic Classifications:
Algebra & Geometry
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
19 Jan 2021 EPSRC Mathematical Sciences Fellowship Interviews January 2021 - Panel B Announced
24 Nov 2020 EPSRC Mathematical Sciences Prioritisation Panel November 2020 Announced
Summary on Grant Application Form
Number Theory is the study of the integers and their arithmetic applications. While problems in Number Theory can be easy to state, their solutions often become extremely intricate. For example, Fermat's Last Theorem - was formulated in the 17th century, yet only resolved in the 1990's.

A fundamental approach in mathematics is to transform a seemingly difficult problem from one area to another, where it becomes tractable or even obvious. A famous example, is Wiles' proof of Fermat's Last Theorem; the key change in perspective transforming a problem about certain arithmetic objects (Galois representations of elliptic curves) into one about analytic objects (modular forms).

This correspondence established by Wiles completing the proof of Fermat's Last Theorem is a very special case of a broad web of predicted correspondences and connections between analysis and arithmetic, collectively known as the Langlands Programme. The Local Langlands Programme is the specialization of the Langlands Programme at a prime number, and this is where the bulk of the research of our project takes place.

The language of the Local Langlands Programme is in a branch of Algebra called Representation Theory, which deals with symmetries of spaces. The Local Langlands Programme is a deep statement that certain fundamental symmetries of finite dimensional spaces which arise in Number Theory can be understood in terms of completely different symmetries of infinite dimensional spaces, and conversely.

The finite and infinite dimensional spaces considered are built on top of the complex numbers. A natural question now arises, why the complex numbers? Is there a more fundamental arithmetic connection hiding behind this? In this project, using explicit constructions of representations, we study integral structures in the Local Langlands Programme and their relation.
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Organisation Website: http://www.shef.ac.uk