Dimer models with boundary

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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.