To kick-start this year's noncommutative geometry (NCG) seminar, I shall give a quick survey of some major recent results. They are mostly related to analytic/geometric aspects of spectral triples and their applications.

In this talk, we will show that any bounded operator on a separable, reflexive, infinite-dimensional Banach space admits a rank-one perturbation which has an invariant subspace of infinite dimension and codimension. In the non-reflexive spaces, we show that the same is true for operators which have non-eigenvalues in the boundary of their spectrum. In the Hilbert space, our methods produce perturbations that are also small in norm, improving on an old result of Brown and Pearcy.

We will also show that if a (norm-closed) algebra A of operators on the Hilbert space has a non-trivial common almost-invariant subspace X (i.e., every member T of A can be perturbed by a finite-rank operator F_T so that X is invariant for T-F_T), then A admits a genuine non-trivial invariant subspace. Time permitting, we will talk about operators having many almost-invariant subspaces. Our principal result here is: if every projection from a masa produces an almost-invariant subspace for the operator T, then T=D+F where D is in the masa and F is finite-rank. This is a finite-rank version of a result of Johnson and Parrott from 1972.

Klaus von Klitzing was awarded a Nobel prize in physics in 1985 for his discovery of the quantum Hall effect in 1980. In this talk I shall first give a quick introduction to the physics of the quantum Hall effect. I shall then explain how quantization of the Hall conductivity at low temperatures can be understood using tools from noncommutative geometry as pioneered by Jean Bellisard. Connes' index theorem relating K-theory and cyclic cohomology plays an important role here. (Note: This Thursday von Klitzing will give a talk at 5:30 PM in Talbot College.)

The category of L-complete modules is an interesting abelian subcategory of R-modules (for a nice ring R) which appears notably in algebraic topology. We will describe different interpretations of this category and some of its features. We will then discuss how completeness interacts with additional structure, such as algebras and lambda-algebras over R.