Mathematics

Real spectral triples with a finite-dimensional Hilbert space are introduced. An example is the fuzzy sphere, which approximates the Riemannian sphere in a certain way (arXiv:1502.05383). Random non-commutative geometries are introduced by integrating over the space of Dirac operators that form a spectral triple with a fixed algebra and Hilbert space. Lisa Glaser and I have investigated the properties of simple cases of this statistical system using Monte Carlo methods (arXiv:1510.01377). Preliminary results indicate that some of the models have a phase transition, with interesting behaviour near the transition.