Graphs in Representation Theory

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.

The impact of this research is mainly in academia and particularly in mathematics, with potentially transformative outcomes in the areas of algebra, geometry, combinatorics and number theory.

The impact generated by this research program will arise in the following ways:

1. This project builds links between the areas of algebra, geometry, combinatorics and number theory. The transfer of methods and ideas between these fields, will enhance the understanding of each of them. Potential beneficiaries of this are researchers in the respective fields and the wider mathematical community.

2. The project has the potential to serve as a catalyst to more collaboration between these research communities.

3. This project will have an impact on the UK's research reputation, in particular in the core areas of the project such as representation theory, matroid theory, cluster theory and algebraic number theory. The international community does take note of the EPSRC fellowships and so a fellowship in these areas will enhance the research reputation of these areas.

4. This project will have a strong impact on the UK mathematics community by establishing new international links and bringing internationally renowned researchers to the UK through the visiting researchers, the two workshops that are part of this project and other new collaboration that this research might generate.

5. An important part of the research in this proposal is on matroids and hypergraphs. These play an important role in computer science, for example in inter-relational databases, in declustering problems which are important to scaling up the performance of parallel databases, in machine learning or in Boolean Satisfiability Problems (SAT). Indeed companies such as Microsoft Research invest and support research in this direction. While we do not expect our research to have a direct application in this area, we are planing to widely disseminate and communicate our results in this area, to keep abreast of the newest developments and if possible push our results to deliver possible applications in these areas.

6. Engaging the wider public in abstract core research activities is crucial for the future support of fundamental core research such as pure mathematics. The impact of core research is often further down the pipeline several steps removed from the initial fundamental research. Nevertheless, this is where the building stones for future breakthroughs in concrete applications are laid. One potential such application could arise from the research in this project related to hypergraphs. Hypergraphs are ubiquitous in many subjects such as computer science (relational databases, greedy algorithm, optimisation) and applied sciences (system modelling in engineering and chemistry). Raising awareness of the possible longterm benefits of fundamental research with the general public will contribute to the understanding and continued support of such research, leading to important socio-economic contributions in the future.
Through the targeted public engagement, facilitated by the attendance of a Royal Society's Residential Communication and Media Skills Workshop, and the ensuing improved dissemination of this research through a general public lecture, radio appearances (building on my experience on a number of appearances on local radio) and press releases (facilitated by the University press office), this research will engage the general public with the importance of fundamental research in mathematics.

7. Through the planned outreach activities, such as the delivery of Master classes to sixth form students, this project will promote mathematics as a STEM subject inspiring a new generation of young people to further their mathematical studies. Waking an appetite for research in Mathematics in the younger generations will contribute to the future scientific base of the country.