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Details of Grant 

EPSRC Reference: EP/P016294/1
Title: Graphs in Representation Theory
Principal Investigator: Schroll, Professor S
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Mathematics
Organisation: University of Leicester
Scheme: EPSRC Fellowship
Starts: 01 May 2017 Ends: 30 April 2022 Value (£): 1,015,963
EPSRC Research Topic Classifications:
Algebra & Geometry
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
24 Jan 2017 EPSRC Mathematical Sciences Fellowship Interviews January 2017 Announced
29 Nov 2016 EPSRC Mathematical Sciences Prioritisation Panel November 2016 Announced
Summary on Grant Application Form
This intra-disciplinary proposal links algebra, combinatorics and number theory through the introduction of new geometric and combinatorial structures. These fields lie at the cutting edge of modern mathematics research and promise potential benefits in applications ranging from theoretical physics to computer science and optimization problems in a wide variety of contexts such as logistics, economics and machine learning.

By introducing special classes of graphs and their generalizations, such as ribbon graphs and hypergraphs, to algebras and their representations, I recently established an exciting link between geometry, combinatorics and non-commutative algebra. This proposal builds and expands upon this new knowledge. More precisely, through novel ideas it will introduce combinatorial objects, the so-called matroids, to catalyse the study of one of the most ubiquitous classes of algebras: wild algebras. Matroids and the hypergraphs that give rise to them are generalizations of graphs that find applications in optimization problems, image clustering and artificial intelligence.

Non-commutative algebra and representation theory in particular is the study of symmetries through the action of collections of linear transformations on vector spaces. A (finite) group is a collection of linear transformations that are invertible. Algebras are more general in that they also model non-invertible processes. Algebras can be divided into two classes: tame and wild. Tame algebras generally have a well-behaved representation theory and the majority of the work in representation theory to date has been devoted to their study. In contrast, there are currently very few tools available to study wild algebras and their representation theory. At the same time, most naturally occurring algebras are wild.

The proposed research introduces new tools for wild algebras in the form of geometric surface models based on novel applications of combinatorial structures including hypergraphs and matroids. Geometry is concerned with the configurations and spatial relations of geometric objects such as points, lines and circles. In modern geometry, such basic geometric objects and their arrangements in space encode complicated structures whose origins arise, for example, from models of the physical world such as string theory, a mathematical model describing the fundamental forces in nature and all forms of matter.

The most basic objects in number theory after integers are fractions of integers, also known as rational numbers. An important open question in number theory is the characterisation of the action of the absolute Galois group, a group based on the rational numbers, on a set of graphs introduced by Grothendieck, called dessins d'enfants. The central related open problem is to find invariants characterizing this action. This research aims to generate new such invariants through the application of the connections of algebra and combinatorics established in the proposed research.

Key Findings
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Potential use in non-academic contexts
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Organisation Website: http://www.le.ac.uk