Approximate Commutators and K-theory

In classical physics, order does not matter: for example, if you measure momentum and then measure position, you get the same answer as if you do the measurements in the opposite order. In quantum mechanics, this is wrong: if you measure momentum before position, you can get a different answer than if you do the measurements the other way around; this is a manifestation of the famous Heisenberg uncertainty principle. Mathematicians call this phenomenon – that order matters – “noncommutativity.” Noncommutative systems are much harder to understand than classical ones. The bulk of this project, which aims to better understand noncommutative systems, concerns the so-called “universal coefficient theorem problem.” On a very rough level, it asks whether it is true that any (reasonable) noncommutative system can be replaced by a classical one, such that doing computations in the classical system gets the same results. Other parts of the project concern direct applications of the same ideas to approximation problems in the study of geometry and symmetry. The project will also involve both undergraduate and graduate students in research. The project intends to use approximation methods to study the so-called universal coefficient theorem (UCT) problem, which asks if any separable nuclear operator algebra (potentially highly noncommutative) is equivalent on the level of K-theory to a commutative one. The recent impressive successes in the Elliott program to classify simple nuclear C*-algebras have thrown this problem into sharp relief: it is now the only extra ingredient needed to get the best possible classification result. The approximation methods should be useful in other areas: two projects building on these ideas study approximation properties of word hyperbolic groups and finite-dimensional models for elliptic differential operators on manifolds. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.