Tag - Non-commutative geometry

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.

In this mini-course I will introduce the universal procedure of topological recursion, both by treating examples and by presenting the general formalism. We will study the classical case of the Hermitian matrix model in detail, which combinatorially corresponds to ribbon graphs, beginning from the loop equations, which correspond to Tutte’s recursion in the combinatorial setting. This will be the starting point to make the connection to free probability, which moreover provides a combinatorial way of exploring the variation of the topological recursion output when applying a symplectic transformation to the input. Apart from the (conjectural) property of symplectic invariance, topological recursion has many other interesting features and, together with its generalizations, has established connections to various domains of mathematics and physics, like intersection theory of the moduli space of curves and integrability. We will explain some of these properties and connections, giving several ideas why this is worth considering, and is the starting or gluing point of an active field of research, and finally hoping to instigate the search of new beautiful connections.