Dimer models with boundary

A dimer model is a graph, meaning a set of nodes connected by edges, drawn on a surface. The nodes of the graph are coloured black and white, and edges may only connect nodes of different colours. Despite this apparently simple definition, a dimer model records an astonishing amount of mathematical and physical data. While dimer models first appeared in statistical mechanics, for example in studying thermodynamical behaviour of liquids having molecules of two different sizes, they have turned out to be useful in a broad range of areas, reappearing later in string theory and algebraic geometry. From an algebraist's point of view, the most important piece of information encoded in a dimer model is the dimer algebra, a collection of paths in the surface that can be multiplied together according to geometrically-motivated rules.

So far, most research in this area has concerned dimer models on closed surfaces, such as the torus (the surface of a doughnut). Many such dimer models are consistent, meaning that they have extremely strong symmetry properties; in technical language, their dimer algebras are 3-Calabi-Yau. This is part of what makes dimer models interesting to string theorists, since 3-Calabi-Yau algebras are closely related to Calabi-Yau manifolds---in many string theoretical models, the four spacetime dimensions of the universe are augmented by an additional six dimensions arising from such a manifold. Studying the properties of consistent dimer models, as well as different ways of detecting consistency from the graph, has led to a great deal of interesting research across mathematics and theoretical physics.

More recently, mathematicians (e.g. Baur, King and Marsh) and physicists (e.g. Franco and collaborators) have been led independently to consider dimer models on surfaces with boundary, such as discs, in the context of cluster algebras and representation theory on the mathematics side, and in various physical problems such as the calculation of scattering amplitudes. There are many natural examples of such dimer models, for example those arising from the study of maximal non-crossing collections and their relationship to Grassmannian cluster algebras. The introduction of a boundary leads to many new phenomena, since much of the data associated to the dimer model, such as its dimer algebra, behaves very differently near the boundary compared to in the interior of the surface. In particular, this different boundary behaviour means that the dimer algebra will not be 3-Calabi-Yau in a strict sense. However, this property is not totally lost, and the dimer algebra still shares many properties with 3-Calabi-Yau algebras.

My recent work gives a precise definition of an 'internally 3-Calabi-Yau algebra', which captures the idea of an algebra being Calabi-Yau in its interior, but with different behaviour at the boundary. This new notion opens up the possibility of extending the many fruitful areas of research on dimer models on closed surfaces into a wider context, and the fellowship intends to exploit this opportunity by developing a theory of consistent dimer models with boundary, the dimer algebras of which are internally 3-Calabi-Yau, and investigating consequences of this symmetry. Moreover, it will address questions that can only arise for dimer models with boundary, such as the problem of how to glue such dimer models together in such a way that consistency is preserved, which is also of interest to physicists, and explore links to the emerging and vibrant mathematical theory of cluster algebras, which are more pronounced in the boundary case.

This fellowship concerns fundamental research in pure mathematics. Such research has historically led to huge impacts beyond academia, although these long-term consequences typically manifest in surprising ways and are highly difficult to predict, particularly at the level of individual research projects. In order to continue to deliver this significant impact, it is important to maintain and strengthen research activity in pure mathematics as a whole, which this fellowship, with its connections to many different areas of mathematics and theoretical physics, will contribute to.

The mathematical theories of dimer models and cluster theory are relatively young, with their study only beginning in earnest in this century, but have already proved to have applications well beyond the problems that they were designed to solve. The theory of cluster algebras in particular is highly significant in this respect; while its development was motivated by fairly specific problems involving canonical bases in Lie theory, connections have since been found to subjects as diverse as complex analysis and mathematical biology. This is made more remarkable by the fact that the subject is only around 20 years old, making it extremely young in mathematical terms.

The appearance of these surprising applications, while for the time being primarily limited to other theoretical subjects, is a characteristic of mathematical techniques which eventually prove to be valuable in more applied settings; for example some recent developments in image processing and text recognition involve the theory of quiver representations, which was developed around 50 years ago as an abstract tool for dealing with very general linear algebra problems, and quickly proved to be a powerful technique in a number of mathematical areas, including those considered in this fellowship. The impact aims of the current proposal are to increase the potential for future non-academic impact of dimer models and cluster theory both by deepening the theoretical understanding of the mathematical techniques involved, in order to better understand why they are so effective, and by 'passing on the baton', by actively disseminating these techniques to researchers in neighbouring fields. In particular, I will organise a workshop in Leeds designed to foster interactions between mathematicians and physicists working in this area. As explained in the Pathways to Impact, I will also conduct exploratory activities to actively investigate further areas in which the results of this fellowship may be applied.

The proposal will have additional impact and benefit to the UK by supporting research areas in which the UK is world-leading, adding to an influential but currently rather small group of UK-based researchers driving work on the links between representation theory and cluster algebras. It will support me at a crucial early career stage, as I work on developing a world-leading research profile, in keeping with EPSRC's strategic priority of securing future talent. The fellowship also brings together techniques and ideas from several existing EPSRC funded projects, enhancing the interconnectedness of EPSRC's portfolio.

I will also aim to deliver impact through the promotion of mathematical sciences via outreach activities, such as masterclasses for pre-university students. Such classes are designed to encourage a new generation to study mathematics at an advanced level, contributing to the UK's skills base. Using ideas central to the research fellowship, I will demonstrate to students the richness and variety of mathematical thinking, beyond rote calculation, by illustrating the way in which translating a problem into different mathematical languages (such as between algebra and geometry) can offer new insights into its solution. In this way I aim, alongside the many other researchers participating in such programmes, to inspire young peoples' interest in the mathematical sciences.