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Bound states, reflection, and topology

Designing electromagnetic media usually involves the detailed specification of material parameters as a function of position. Topological methods are different, and a device can be largely insensitive to the local details of the material. In this presentation I shall discuss some new applications of topology to material design, showing that the Atiyah-Singer index theorem determines the existence of modes in inhomogeneous media via the material properties at the edge of the device, and can also be applied to design reftectionless non-Hermitian media.