PhD Thesis Defence Public Lecture (Math) - Nathan Pagliaroli

In the  Noncommutative Geometric setting of spectral triples Dirac operators take the place of metrics via Connes’ distance formula.  In theories of quantum gravity one is often interested in constructing integrals over metrics/topologies. Inspired by this in 2016 Barrett and Glaser constructed toy models of quantum gravity on finite dimensional noncommutative spaces.  One fixes the information of the spectral triple except the Dirac operator which is assigned a probability distribution. Such a combination is referred to as a Dirac ensemble. The finite dimension of the spectral triple reduces integrals over these spaces of Dirac operators to matrix integrals. In a series of papers with M. Khalkhali and H. Hessam, we have rigorously studied Dirac ensembles using techniques from random matrix theory. Physical motivation aside, Dirac ensembles display many interesting mathematical properties, such as Eynard-Orantin topological recursion, spectral phase transitions, and limit laws.