A cosmological correlator is an Euler integral, associated to a graph G, which encodes information about the state of the early universe. Evaluation of these integrals is extremely challenging, even in simple cases. However, it turns out the integrand can be identified with the so-called canonical form of the cosmological polytope, revealing a rich combinatorial structure and allowing the application of techniques from commutative algebra. I'll sketch my contribution to this story and advertise the fledgling field of positive geometry, which seeks to generalize the notion of canonical forms to geometric objects more exotic than polytopes.

Dirac ensembles lie at the intersection of random matrix theory and noncommutative geometry. In particular, we look at the (0,1) fuzzy Dirac ensemble with fermion in the case where the fermion mass is positive. We give an introduction to random matrix models, both analytic and formal, including the relationship between matrix integrals and generating functions of maps in the formal case. We also motivate and introduce the concept of a spectral triple from noncommutative geometry. One of these applications is Dirac ensembles, consisting of a certain finite spectral triple which describes a toy model of quantum gravity. We describe Dirac ensembles and how they lead to hermitian matrix models, giving special attention to the (0,1) fuzzy Dirac ensemble with a fermion. Loop equations for this model can be found through complex analytic techniques, and one allows for a perturbative solution of the moments. In the case where the potential is Gaussian, an exact solution for moments can be obtained using complex analytic techniques and elliptic integrals.