  Framed combinatorial topology is a novel theory describing combinatorial
phenomena arising at the intersection of stratified topology, singularity
theory, and higher algebra. The theory synthesizes elements of classical
combinatorial topology with a new combinatorial approach to framings. The
resulting notion of framed combinatorial spaces has unexpectedly good behavior
when compared to classical, nonframed combinatorial notions of space. In
discussing this behavior and its contrast with that of classical structures, we
emphasize two broad themes, computability in combinatorial topology and
combinatorializability of topological phenomena. The first theme of
computability concerns whether certain combinatorial structures can be
algorithmically recognized and classified. The second theme of
combinatorializability concerns whether certain topological structures can be
faithfully represented by a discrete structure. Combining these themes, we will
find that in the context of framed combinatorial topology we can overcome a set
of fundamental classical obstructions to the computable combinatorial
representation of topological phenomena.
