The Application of Topology to Waves: An Introduction

In this chapter we give an introduction to the mathematics of topology, and its application to problems in waveWaves physics. Starting from the theory of closed surfaces, we review the origins of the Chern number. After showing that this number counts the critical points of a complex tangent vector field on a surface, we then show that this quantity arises naturally when calculating the dispersionDispersion of modes in any linear system. We conclude by giving examples of the application of topology to find one-way propagating interface modes in both continuous and periodicPeriodic materials.