This project involves computing some Feynman type integrals in some finite dimensional discrete settings.

The motivation comes from a toy model for quantum gravity. It is known that the fabric of spacetime in very short distances is not classical anymore and should be replaced by a hitherto mysterious quantum spacetime. At the same time one needs to take into account, that is to integrate over, all possible geometries. Quantum gravity is ''terra incognita", but in the present model different geometries, that is different metrics, can be parametrized by spaces of self-adjoint matrices. All integrals are finite dimensional and well defined. One goal is to understand if there are any kind of universal laws governing the distribution of eigenvalues as the size of matrices grow. While there are some similarities with random matrix theory, the nature of the current project is quite different and the subject is still in its infancy. Most of the integrals will be computed by computer simulation and Monte Carlo methods, but a theoretical understanding of them would be very important. A good undergraduate math education is enough to handle this project.