The spectrum problem for noncommutative rings and algebras

The spectrum of a commutative algebra is a topological space, assigned in such a way that homomorphisms between algebras correspond to continuous functions between spaces. Since the spectrum lends geometric intuition to the study of commutative algebra, it seems natural to follow the tradition of noncommutative geometry and ask: what might be a useful spectrum for noncommutative algebras? I will explain an unexpected connection between this problem and Kochen-Specker contextuality in quantum physics, showing that the physical theory provides an obstruction to this purely mathematical question and providing a preliminary report on work (joint with M. Ben-Zvi, A. Chirvasitu, and A. Ma) toward algebraic variants of the Kochen-Specker obstruction. Then I will discuss some progress (joint with Chris Heunen) toward a positive result in the form a correspondence between certain operator algebras and "quantum Boolean algebras" called active lattices.