Poincare told us more than a century ago that periodic orbits `` ..are the only breach through which we can penetrate into a place which up to now has been reputed to be inaccessible.'' Our group has taken Poincare's advice seriously, and has constructed a discrete classification for low-dimensional strange attractors. This discrete classification depends on the unstable periodic orbits in a strange attractor. The classification consists of a hierarchy with four levels of structure: 1. basis sets of orbits, which force all the unstable periodic orbits in a strange attractor (braid theory); 2. Branched manifolds, which organize all the unstable periodic orbits in a strange attractor in a unique way (Birman-Williams theorem); 3. bounding tori, which organize branched manifolds in the same way that branched manifolds organize periodic orbits (Euler Index and Poincar'e Hopf index theorem); 4. embeddings of bounding tori in R^3 (more braids). To extend this classification to higher dimensional strange attractors, we need to learn how to determine orbit organization in higher dimensional spaces. A program for extending this work to higher dimensions will be outlined, indicating where some useful contribution can be made: linking numbers beyond gauss; inertial manifold constructions; structure theory for flows (reducible, fully reducible, irredicuble); representation theory for branched manifolds; germs and unfoldings of mappings; ranks of singularities; complexification and real forms; covers and images; an inverse for Cartan's theorem for Lie groups; universal image dynamical systems.