The feasibility problem associated with nonempty closed convex sets $A$ and $B$ is to find some $x\in A \cap B$. Projection algorithms in general aim to compute such a point.

These algorithms play key roles in optimization and have many applications outside mathematics - for example in medical imaging.

Until recently convergence results were only available in the setting of linear spaces (more particularly, Hilbert spaces) and where the two sets are closed and convex.

The extension into geodesic metric spaces allows their use in spaces where there is no natural linear structure, which is the case for instance in tree spaces, state spaces, phylogenomics and configuration spaces for robotic movements.

After reviewing the pertinent aspects of CAT(0) spaces introduced in Part I, including results for von Neumann's alternating projection method, we will focus on the Douglas-Rachford algorithm, in CAT(0) spaces. Two situations arise; spaces with constant curvature and those with non-constant curvature. A prototypical space of the later kind will be introduced and the behavior of the Douglas-Rachford algorithm within it examined.

Geodesic metric spaces provide a setting in which we can develop much of nonlinear, and in particular convex, analysis in the absence of any natural linear structure. For instance, in a state space it often makes sense to speak of the distance between two states, or even a chain of connecting intermediate states, whereas the addition of two states makes no sense at all.

We will survey the basic theory of geodesic metric spaces, and in particular Gromov's so called CAT($\kappa$) spaces. And if there is time (otherwise in a later talk), we will examine some recent results concerning alternating projection type methods, principally the Douglas--Rachford algorithm, for solving the two set feasibility problem in such spaces.