EPSRC Reference: 
EP/W007509/1 
Title: 
Combinatorial Representation Theory: Discovering the Interfaces of Algebra with Geometry and Topology 
Principal Investigator: 
Baur, Professor K 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Pure Mathematics 
Organisation: 
University of Leeds 
Scheme: 
Programme Grants 
Starts: 
01 August 2022 
Ends: 
31 July 2027 
Value (£): 
2,554,972

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Logic & Combinatorics 
Mathematical Physics 


EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
A fundamental, and often successful, way of studying an abstract mathematical object is to consider methods of representing it in another, more concrete object. This is a powerful idea, and recent progress in algebraic representation theory and related areas has given rise to strong opportunities for the transformation of other fields. In particular, geometric and combinatorial phenomena initially specific to representation theory have emerged in many other fields, leading to effective new techniques and applications. Our team is at the forefront of these developments. The PI and the five CoIs have contributed to major advances in the past decade, with their expertise ranging from algebra, geometry, and topology to mathematical physics. This provides new ways to link algebra and geometry & topology. Examples include the categorification of the Grassmannian cluster structure, the McKay correspondence for reflection groups, the lifting of Lietheoretic techniques to 2dimensional category theory, with applications to topological physics, and the derivation of decomposition matrices of Brauer algebras from generalised Lie geometry. In all cases, the medium for interpolating between the theories is an emergent geometrical property which is not well understood. For the advancement of research, there is a strong need for explaining these phenomena and placing them in an encompassing novel paradigm. Our proposal hence seeks to understand and investigate relations between very different areas, and so to push on from there in a more systematic framework. This aim would benefit from a broad, holistic view of representation theory, embracing Lie theory, algebraic geometry, low dimensional topology and mathematical physics. Our team in Leeds is uniquely qualified to pursue this programme. Together with specialist collaboration of many mathematicians at our international partner institutions, we will address the current challenges, provide solutions to open questions and develop applications by establishing bridging to other fields. We are in a position to embrace the perspectives of both pure and applicationdriven mathematics, and with the potential, in the long term, for serving the needs of physical sciences, life sciences and engineering. This unification of perspectives requires a programmelevel research structure and algebra is the right core platform for such an ambitious venture. Thus our proposal will push forward the mathematical stateoftheart and will shape the future directions in the areas we touch upon.

Key Findings 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Potential use in nonacademic contexts 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Impacts 
Description 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk 
Summary 

Date Materialised 


Sectors submitted by the Researcher 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Project URL: 

Further Information: 

Organisation Website: 
http://www.leeds.ac.uk 