Colloqium: Operators, Alegebras, and Spaces

Abstract: The study of spaces through global invariants, local geometric features, and the interrelations between these methods is a subject that goes back to the greats such as H. Poincaré, E. Cartan, and S.S. Chern. It was discovered by M.F. Atiyah and I.M. Singer that these classical studies typically involved a natural Fredholm operator which has baked into it the global invariants as well as the local geometric information. 

This story has blossomed into a subject, Index Theory, and the broad aim of the subject is to understand interrelations between global invariants on a space, operators defined on spaces, and geometric properties of spaces. From the subjects inception, the quest to understand these various bridges has been guided by difficult questions in the field, many of which require the introduction of noncommutative algebras for their resolution. A few examples will be provided, and we will end with a discussion on recent developments connecting noncommutative harmonic analysis to index theory and our current work here. This work was joint with Andres Franco Valiente.