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Dr Barnes studies algebraic topology, specifically stable homotopy theory, usually with an equivariant or monoidal flavour. The method of algebraic topology is to assign algebraic objects (such as vector spaces) to topological shapes in a homotopy invariant manner. When these shapes are chosen to have some group of symmetries, the vector spaces assigned to them become representations of the symmetry group. Thus his interests include stable homotopy theory, homotopy functor calculus, homotopical algebra, model categories and equivariant homotopy theory.