THE MAGNETIC SPECTRAL TRIPLE: APPLICATIONS AND OPEN QUESTIONS

Since the early works by Bellissard, noncommutative geometry has proved to be an excellent tool for the study of the topological phases of matter in general, and for the analysis of the quantum Hall effect in particular. The central object of Bellissard's noncommutative geometry of the quantum Hall effect is a spectral triple designed to deal with tight-binding operators. In this talk, we will present a new spectral triple suitable to treat continuous magnetic operators. We will show how the quantum Hall effect in this continuous setting can be described within the new framework. Certain possible new applications, along with some related open questions, will also be presented. (Joint work with F. Belmonte and M. Sandoval.)