Topological fermion systems provide a general framework for desribing and analyzing non-smooth geometries. They can also be used to describe "quantum spaces" or "quantum space-times" as considered in quantum gravity. Moreover, they set the stage for the so-called fermionic projector formulation of relativistic quantum field theory.
The aim of the talk is to give a simple introduction, with an emphasis on conceptual issues. Starting from a collection of functions on (which can be thought of as Schrödinger wave functions), we ask the question whether the geometry of the Euclidean space is encoded in these functions. Bringing this question into a precise mathematical form leads us to the abstract definition of topological fermion systems. This definition will be illustrated by the examples of vector fields on the sphere, a vector bundle over a manifold, and a lattice system. As an example motivated from physics, we briefly consider Dirac spinors on a globally hyperbolic Lorentzian manifold and introduce the setting of causal fermion systems. The inherent geometric and analytic structures on a topological fermion system are introduced and explained. A brief outlook on the applications to quantum field theory is given.
