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Abstract

The operator algebras that I work with are C*-algebras: norm-closed *-algebras of bounded linear operators on Hilbert space. These arose, via the genius of von Neumann, from Heisenberg, Born and Jordan’s “matrix mechanics” description of quantum observables. These days they retain active connections to mathematical physics, but are a major field of mathematical investigation in their own right and provide a very useful setting for linear representations of algebraic, dynamical, topological and combinatorial data. Unfortunately, they are very hard to visualise in the abstract. I will give an overview of my work on finding nice “coordinate systems” for C*-algebras that can help us to visualise and study them, and some good ways of building examples from intuitive and easy-to-manipulate combinatorial data, and try to explain how these two ideas talk to each other.