On special sets and their geometry

I will discuss a very general definition of a special subvariety inspired by recent works on those for Shimura varieties by Pila, Ullmo, Yafaev and others. The main theorem states that the general special subvarieties are exactly closed irreducible subsets of a certain $1$-based Zariski structure, so are essentially classifiable. This allows to formulate a general form of a Schanuel-type conjecture and, correspondingly, a Diophantine conjecture on atypical intersections.