Abstract: Matrix integrals play an important role in random matrix theory, 2d quantum gravity, and the topology of moduli spaces of Riemann surfaces. In these lectures we shall look at a perturbative expansion for matrix integrals via Feynman graphical methods.

Abstract: We introduce the first example of a higher inductive type, S^1, for which we prove a universal property. Then after discussing the interval I, we move on to spheres. We finish with finite CW complexes.

Abstract: In this talk I will discuss how to use ideas from the theory of metric spaces of negative curvature to understand the behavior of holomorphic maps between bounded domains in complex Euclidean space. Every bounded domain has an metric, called the Kobayashi metric, which is distance non-increasing with respect to holomorphic maps. Moreover, this metric often satisfies well-known negative curvature type conditions (for instance, Gromov hyperbolicity or visibility) and one can then use this negative curvature to understand the behavior of holomorphic maps. I will discuss the domains where the Kobayashi metric satisfies negative curvature type conditions and how to use these conditions to prove new results. Some of this is joint work with Gautam Bharali.