Diverse key phenomena in pure and applied mathematics arise from the fact that the familiar commutative law for numbers (the formula ab=ba) fails for more general quantities. Notably, it fails for the operators describing the position and the momentum in quantum mechanics, and the size of their commutator is the source of Heisenberg's uncertainty principle. More general commutators in the functional calculus of these operators are key to other important phenomena such as factorisation of functions, compensated compactness in partial differential equations, and quantum differentiability. Over the past few years, this theory has greatly expanded into new directions, but at the same time become more scattered. The aim of this project is to develop a general framework that covers, unifies, and simplifies many of these previous results with far-reaching extensions.

2026

2031