Linking symmetry, crystallography, topology, and fundamental physics in crystalline solids

In this chapter, we briefly introduce the evolution of symmetry as a mathematical concept applied to physical systems and lay the mathematical groundwork for discussion of topological physics. We explain how topological phases, like the Berry phase, can be obtained from a gauge symmetry of a quantum system. Also, we introduce numerical tools (e.g., Chern numbers, Wilson loops) for topological analysis of chemical solids based on the crystal structure and corresponding electronic structure