The importance of modern topology in proving robustness of physics models and experiments, has been recognised by several recent Nobel prizes. As a paradigmatic example, I will outline the mathematics behind the topological insulator phenomenon. Its qualitative spectral features turn out to be in duality with hole-counting in abstract classifying spaces, and so enjoy “topological protection”. Thus topological materials are physical manifestations of a beautiful unity of geometry, topology, algebra, and analysis lying at the heart of index theory.
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