One of the cornerstones of condensed matter physics, the description of wave functions on periodic lattices in terms of energy bands of Bloch states, serves as the unifying thread in this thesis. This description is often referred to as band theory. …

Topological band theory studies the behavior of non-interacting electrons in solids that is protected by topological invariants and is therefore robust against system perturbations. Among many topics that are important for characterizing topological …

The physics of negatively curved (hyperbolic) geometry has experienced a large resurgence of interest in recent years, both theoretically and experimentally. Although hyperbolic spaces constitute a fundamental ingredient in theoretical studies of …

Topological semimetals are characterized by topologically protected band-structure nodes. One prominent example is the Weyl semimetal, characterized by Weyl points carrying topological Chern numbers. In this dissertation, we explore the topology of …