October 29, 2001. S. V. Ivanov. 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 R), and tell about intriguing nonlinear constraints arising in another variant (which corresponds to ellipticity over Z).

List of talks at previous sessions of the seminar.