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ManifoldsA manifold is a mathematical space which is locally like the usual coordinate space of some fixed dimension, known as Euclidean space. The earth itself is an example of a two-dimensional manifold: maps exploit the fact that, if you don't go too far from a given point, the earth behaves pretty much like a plane. The earth could be covered entirely by small maps making use of this local planar structure, although an accurate map of the entire world must be a globe. A one-dimensional manifold, such as a circle, looks locally like a line, and there are manifolds of higher dimension as well. Manifolds play an important role in mathematics because there are so many examples and because they can be studied using the properties of Euclidean space. SymmetriesThe symmetries of an object are the transformations which leave it looking exactly as it was before, such as a rotation of a square through a quarter of a revolution. Two symmetries can always be composed to give a third. This, together with other properties, gives the set of symmetries of any object the structure of a group. While there are only 8 symmetries of a square, the circle has infinitely many symmetries, forming a one-dimensional manifold. The manifold and group structures are compatible, and such an object is known as a Lie group. Lie groups often arise as certain symmetry groups known as gauge groups in physics as well.Lie groupsIn the 1870s, Sophus Lie studied Lie groups by linearizing them, to obtain simpler spaces known as Lie algebras. Lie algebras can be studied using more elementary mathematical notions from linear algebra, which is the study of vector spaces (such as Euclidean spaces) and their linear transformations. This and other developments led to the modern subject of Lie theory in which Lie algebras are studied via their representation theory: each element of the Lie algebra is represented by a transformation of a vector space. Such transformations can be represented more concretely via matrices, which are rectangular arrays of numbers, provided a basis is chosen (essentially a choice of coordinates).Quantum groupsIn 1985, deformed versions of the classical Lie algebras, or quantum groups, were introduced by Drinfeld and Jimbo, and this led to a revolution in the way that Lie algebras were studied. In particular, Kashiwara and Lusztig introduced the canonical basis in 1990, with beautiful properties. It simultaneously gives rise to bases for all representations for the Lie algebra (of a certain kind). The canonical basis has applications to other fields as well, such as the representation theory of affine Hecke algebras.The projectAttempts to describe the canonical basis explicitly have led to a lot of interesting mathematics, including the cluster algebras of Fomin and Zelevinsky, introduced to model its multiplicative properties. Cluster categories and cluster-tilting theory were introduced in order to understand cluster algebras. These objects were defined using representations of quivers: a quiver is a graph with oriented edges and a representation is a vector space for each vertex and a transformation for each edge. As well as giving insight into the canonical basis, cluster-tilting theory has rapidly become a powerful tool for the study of representations of quivers.This model of the canonical basis is still incomplete and work on the proposed project will help towards a better picture via quantum cluster algebras. The project will also work on developing cluster-tilting theory from the perspective of representations of quivers by describing its combinatorial properties and generalising it to a wider context in several directions. The project will also develop stronger connections to the diagram algebras arising in statistical mechanics.
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