Consider the following question: given a triangulated manifold $M$ compute its characteristic classes from the discrete data of its triangulation. This question is easily answerable for some characteristic classes (Euler, Stiefel-Whitney), but revealed to be very difficult for Pontryagin classes: the problem was tackled in the 1970's in articles by Gabrielov, Gelfand, MacPherson and others, and it heavily influenced algebraic topology at the time. Until 2004, there was no reasonably computable formula even for the first Pontryagin class. Another way to look at computing characteristic classes combinatorially is the following: consider a triangulated bundle (with compact fiber) in general, and compute characteristic classes given a triangulation of the bundle with a simplicial projection map. Known formulas are based on finding the cocycle corresponding to the cohomology class by averaging over "local" cycles.

I will present the history of the problem, some recent results and a lucky example of a beautiful application to minimal triangulations of projective planes.