The next lecture in this series is here.
Our starting point is a spectral approach to geometry, starting with the simple question ’can one hear the shape of a drum?’ This was phrased by Mark Kac in the 1960s, and led to many developments in spectral geometry. For us, it is the motivation for considering spectral triples, which is the key technical device used to describe non-commutative Riemannian spin manifolds. We will give many motivating examples, and also explain how gauge symmetries naturally arise in this context. The connection to the other main theme of the workshop is found via the spectral action principle. It allows for a derivation of an action functional from any given spectral triple. This includes the Hermitian matrix model, but more interesting matrix models appear beyond. We will consider some recent developments for such models by deriving a perturbative series expansion for the spectral action.
This video was produced by the University of Münster, and forms part of the workshop Non-commutative geometry meets topological recursion.