There is a long history of attacking problems involving nonabelian Lie groups by reducing to a maximal abelian subgroup. It has been understood in the last decade that

1. the exact WKB method for studying linear ODEs,
2. the computation of classical Chern-Simons invariants of flat connections,
3. the study of some link invariants, such as the Jones polynomial,

can all be understood as aspects of this general idea. I will describe this point of view, trying to emphasize the common features of all three problems, and (briefly) their common origin in supersymmetric quantum field theory. Parts of the talk are a report of joint work with Dan Freed, Davide Gaiotto, Greg Moore, Lotte Hollands, and Fei Yan.