Ampleness up to avoidance

In the first half of the talk I will review Gromov's work on convex integration for open differential relations. I will put particular emphasis on comparing various flavours of ampleness and, in particular, I will note that the different flavours require slightly different implementations of convex integration.

In the second half of the talk I will explain some recent joint work with F.J. Martínez Aguinaga: We define a notion of "ampleness up to avoidance" and we prove that convex integration applies to open relations satisfying it. Our main example is the differential relation defining hyperbolic (4,6)-distributions: it is not ample in the classic sense, but it is ample up to avoidance and thus abides by the h-principle.