Mathematics Graduate Students Organization

Seminar:

Date: November 5

Many commonly encountered mathematical objects -- for example, probability spaces, topological spaces, and smooth manifolds -- are studied by considering appropriate commutative algebras of functions on those spaces. In operator algebras, we ``quantize" these spaces by replacing commutative algebras by noncommutative algebras with the same or analogous properties. For this reason, the study of operator algebras is aptly nicknamed ``noncommutative mathematics." In this talk, we'll take a fast and loose look at ordinary matrix algebras from a slightly different angle that highlights the general strategy of noncommutative mathematics. This point of view leads naturally to constructions of two ``noncommutative spaces," the noncommutative torus and hyperfinite $II_1$ factor -- perhaps the two most important examples in operator algebras. With these examples in hand, we will obtain a closer look at an object of current interest in the field -- the maximal abelian subalgebra -- which lies at the interface of operator algebras, dynamics, and geometric group theory. Some undergraduate algebra and advanced calculus are the only background required.