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Explore edge states in periodic and aperiodic structures through this mathematical lecture that examines solutions of energy conserving wave equations which remain bounded and oscillatory along line defects while localizing transversely. Delve into edge states within systems exhibiting honeycomb symmetry, particularly those found in two-dimensional quantum materials like graphene and their metamaterial counterparts. Begin with the analysis of "rational edges" - line defects such as domain walls or sharp terminations that align with period lattice vectors. Progress to cutting-edge research conducted with P. Amenoagbadji on edge states that localize transverse to irrational edges, representing the latest developments in this field. Gain insights into wave propagation phenomena in guiding structures through this comprehensive mathematical treatment that bridges theoretical concepts with practical applications in quantum materials and metamaterials.
