The h-principle is the subfield of differential topology that studies the classification of geometric structures on smooth manifolds, up to homotopy. Since its birth, it has played a crucial role in foliation theory, contact and symplectic topology, and the theory of positive scalar curvature, among other areas.

Its origins can be traced to the study of immersions, where a celebrated result of Smale and Hirsch states that the space of immersions is homotopy equivalent to the space of so-called "formal immersions". Formal immersions are algebraic topological objects, and the space thereof can be studied using obstruction theory.

One may ask whether a similar approach can be pursued in order to study instead embeddings. The key difference between the two setups is the "non-locality" of the latter: While the immersion condition amounts to checking full rank of the differential at each individual point, the embedding condition additionally involves the injectivity of every pair of points. This distinction turns out to be profound, and makes the study of embeddings much more difficult.

Still, work of Goodwillie-Klein-Weiss, spanning roughly the period 1990-2015, establishes that the space of embeddings is indeed equivalent to a space of "formal embeddings", as long as the source manifold has codimension at least 3 compared to the target. This is a truly remarkable result, which involves many deep ideas from differential topology and homotopy theory.

The goal of the talk will be to explain this story in some detail. I will do so from a biased perspective: together with Aaron Gootjes-Dreesbach and Bas de Pooter we are working on a long-term programme to extend the work of Goodwillie-Klein-Weiss to a general class of problems involving non-local differential constraints. I will therefore also report on the results we have so far and what remains to be done.