While combinatorial structures are often investigated for their intrinsic beauty, problems of a combinatorial nature arise in many other areas of pure mathematics, notably in group theory, probability, topology, algebraic geometry, geometry and number theory. Bristol has particular strengths in advancing interactions of combinatorics with the latter three areas, specifically in the study of algebraic varieties through their combinatorial counterparts like matroids, hyperplane arrangements, polytopes and lattices; incidence geometry and geometric combinatorics; and Szemeredi-type problems concerning arithmetic structures in subsets of the integers and the primes.