Wilson Loops, Wyckoff Positions, and Wannier Functions: New Developments in Stable and Fragile Topology

The interplay of topology and geometry has been -- and continues to be -- a rich area of study for condensed matter physics. Recently, we have realized that spatial symmetries allow for the stabilization of topological phases much more exotic than those that can be found with time-reversal symmetry alone. Examples include topological crystalline insulators, "hourglass Fermion" phases, and Dirac and double-Weyl semimetals. In this talk, I will review recent developments in the theory of band representations which highlight the role of Wannier functions and holonomy in explaining the origins of topological crystalline behavior. I will show how this relates to several new ideas, such as symmetry indicators, topological phases with high co-dimension boundary states, and the "fragile" topology of isolated groups of bands. Finally, I will discuss how non-symmorphic symmetries can protect novel topological surface states, which can be diagnosed through the holonomy of Bloch functions.