Ph.D. Public Lecture (Math) - Jeremy Gamble

Title: Expansions of the Spectral Action

Abstract: In noncommutative geometry, spectral triples play a central role as noncommutative spaces, as best exemplified by the example of the Dirac operator on a Riemannian spin manifold. We will give some basic definitions and examples about spectral triples, and describe the inner fluctuations of a spectral triple and how they arise naturally from Morita equivalence. The spectral action principle of Connes and Chamseddine will then be introduced, with the inner fluctuations playing a key role as gauge fields in the theory described by the spectral action. We will then discuss some of the existing literature on expanding this spectral action, with a particular emphasis on the work of van Suijlekom and van Nuland. We will see that their expansion can be nicely expressed in terms of integrals over cyclic cocycles, and we will describe how these cocycles appear.