Polyhedral surfaces with prescribed set of face directions and ellipticity of area functionals (joint work with D. Burago)

Let $S$ be a finite set of oriented $k$-dimensional linear subspaces in the $n$-dimensional Euclidean space. Under what conditions there exist a $k$-dimensional piecewise-linear oriented surface $M$, almost all whose faces (that is, all faces except a set of arbitrarily small area) are parallel to elements of $S$, and such that the boundary of $M$ is contained in a $k$-dimensional subspace? This question is a dual problem to an old question from geometric measure theory, namely to the problem of characterization of Almgren's elliptic $k$-area functionals. Recently relations with other fields were also discovered. The problem is nontrivial in dimensions starting from $n=4$ and $k=2$.
In the talk I will explain where this problem came from, what it is related to, the solution for the above variant of the formulation (which corresponds to the so-called ellipticity over $\mathbf R$), and tell about intriguing nonlinear constraints arising in another variant (which corresponds to ellipticity over $\mathbf Z$).