Homological properties of quantum cluster algebras

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This project aims to explore the different types of behaviour of a family of algebras from a homological and geometric perspective. The algebras in question, graded quantum cluster algebras, have a rich combinatorial structure but many questions about them from other points of view, such as their ring-theoretic properties, are not well understood. For example, recent work of Booker-Price on rank 3 cluster algebras has found a variety of possible behaviours for the growth of these graded algebras, and shows that these behaviours correspond to combinatorial properties of the cluster data. The first part of the project will explore this in more generality, so that when we have a better understanding of the range of behaviours, we will be able to identify families having - or not having - desirable homological properties such as being AS-Gorenstein or AS-regular. When we have one of these homological properties, we may then apply the machinery of noncommutative geometry. Moving into this realm of noncommutative geometry opens up the possibility of studying the representation theory of quantum cluster algebras - something that has not been done before. There are too many representations to study them individually but instead one can hope to show that they are parameterised by nice geometric objects, such as curves or surfaces in projective space.