Phonons dominate a wide range of material properties, from the speed of sound to heat transport; drive phenomena like temperature induced structural phase transitions; and their coupling to electrons leads to emergent states of matter such as superconductivity. Calculations of phonons using density functional theory in its semilocal approximations yield results in good agreement with experiment for many materials and phenomena. Despite these successes, in this talk I argue that there is still much to be learnt from looking at phonons afresh.
First, I will describe our recent efforts at calculating phonons with methods beyond semilocal density functional theory, using both hybrid functionals and dynamical mean field theory. Second, I will describe how we can understand phonon dispersions from the point of view of topology, and suggest that topological phonons may be ubiquitous in materials.