This unit covers the basics of Geodesic metric spaces, beginning with generalizing notions from Riemannian geometry into general metric spaces, defining the model spaces which we can later use as comparison spaces for arbitrary metric spaces, working through various constructions of classical hyperbolic space, defining polyhedral complexes, and finally introducing various now classicial results on metric spaces equipped with group actions.

This unit covers the basics of CAT(k) geometry, beginning with its definition, the consequences of the condition (especially for k<=0), the Cartan-Hadamard theorem and Gromov's link conditon for polyhedral complexes, properties of group actions on CAT(0) spaces, the boundaries of such spaces, and constructions on these spaces that preserve desirable metric and group-theoretic properties.