This is a quickly emerging field of research in which geometrical and topological ideas are exploited to project and control the behavior of light. Drawing inspiration from the discovery of the quantum hall effects and topological insulators in condensed matter, recent advances have shown how to engineer analogous effects also for photons, leading to remarkable phenomena like the robust unidirectional propagation of light, which hold great promise for applications.
A topological invariant is a quantity that's unaffected by continuous deformations of the object. For example, a closed 2D surface of any finite 3D object are often characterized by the genus g, which counts the number of holes in the object. Thus, a sphere has a genus of g = 0, and a torus has a genus of g = 1; these two objects cannot transform continuously into each other: Any transformation from a sphere to a torus necessarily involves some discontinuity at which a hole is created; such topological phase transitions are accompanied by a stepwise (quantized) change in a topological invariant.
Under certain conditions, topological invariants are often utilized to describe band structures of periodic crystalline materials. In this context, continuous transformations are those that preserve all symmetries of the system and don't close the bandgap, and a topological phase change requires the bandgap to shut and re-open.
Boundaries between two materials characterized by distinct topological invariants necessarily host special gapless states localized to the boundary.