The basic relationship in algebraic geometry is between a variety and its ring of functions. Arguably a similarly basic relationship in quantum mechanics is between a state space and its algebra of observables. In what sense is algebraic geometry a "classical" (i.e. commutative) phenomenon? How does intuition from quantum mechanics influence how one should think about algebraic geometry and vice versa? What modern areas of research study their interaction?
(This question is loosely inspired by a similar question about representation theory.)