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Combinatorics is concerned with arrangements of discrete objects according to constraints and the study of discrete structures such as graphs. Large graphs underpin many aspects of data science and can be used to model networks.

Lancaster’s research interests in combinatorics are diverse and closely connected to each of the other pure mathematics research themes in the department.

Additive combinatorics explores how to count arithmetic configurations within unstructured sets of integers - those sets for which we have only combinatorial information, such as their density. Techniques in this field have been developed to tease out the latent structure present in such sets, separating it from random 'noise'. Resolving these combinatorial problems can then feed back into more classical problems of number theory, such as detecting arithmetic structures in the primes.

As combinatorial structures provide convenient means of indexing words in noncommuting variables, many problems in probability and analysis can be approached using methods of diagrammatic calculus. Such methods provide a fundamental bridge between combinatorics and algebra, on the one hand, and probability and analysis, on the other. A well-known example is the moment method, through which the Catalan numbers, ubiquitous in combinatorics, are interpreted as the moments of the semicircle law, the analogue of the Gaussian random variable in random matrix theory. Research in our combinatorics group is closely connected to classical and noncommutative probability, including random matrix theory, and mathematical physics.

Measurable combinatorics concerns extensions of combinatorial theorems in the context of Borel graphs with applications including classical geometrical questions. There is also interest in sparse and dense graph limits where, with a suitable metric imposed on the set of finite graphs, limits of convergent graph sequences are studied.

Matroids are a mathematical structure that extends the notion of linear independence of a set of vectors. They have numerous important applications, for example, in Operational Research and Combinatorial Optimisation. Our research in matroid theory concentrates on real world occurrences of the 2 fundamental motivating examples of matroids: row matroids of matrices and matroids defined on graphs.

Combinatorial aspects of geometric rigidity theory (and its applications in fields such as engineering and robotics) such as recursive constructions of classes of (coloured) sparse graphs. This topic underlies the flexibility and rigidity properties of geometric structures, such as finite and infinite bar-joint frameworks in finite dimensional spaces. Further geometric aspects of combinatorics such as polyhedral scene analysis, equidissection and triangulation problems, and the rich interactions between discrete geometry, combinatorics and symmetry.

Many aspects of mathematics share the same underlying classifications in terms of combinatorics such as Coxeter combinatorics and Fuss-Catalan combinatorics. These underlying combinatorial similarities often indicate deeper connections, e.g. the ubiquitous "finite type" classifications in terms of Dynkin diagrams. In representation theory of algebras, combinatorics is used extensively to get concrete understanding of abstract objects. For example, certain algebras can be associated to ribbon graphs, which in turn can be embedded in surfaces, and directed graphs may be used to encode the multiplication in noncommutative algebras.

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