Crystalline symmetry in topological materials

In this talk I will show how we can extract information encoded in the crystalline symmetry representations of solid state systems in order to understand how topological bands can be formed. In particular I will show in two examples how we can explore the sub- and super-group structures of symmetry representations to characterize and engineer topological phases. First, we will understand the unreasonable prevalence of topological phenomena in bismuth and related compounds; and second we will see how spatial defects can make a topologically trivial system host novel embedded topological phases. General consequences and extensions of this approach will be discussed.