What's new for Jacobian determinants and rank-one connections

Geometric analysis of Sobolev mappings (weakly differentiable deformations) of Euclidean domains or Riemannian n-manifolds will continue to enhance mathematical insights into several applied fields: nonlinear elasticity, material science, crystals, and so forth. In this challenge, there is an important place for the Jacobian determinant of the gradient matrix, its sub-determinants and other null-Lagrangians. We shall emphasize the fundamental role of the Orlicz-Sobolev spaces to capture fine properties of those differential expressions. This brings us closer to PDEs (Hodge theory and very weak solutions) and modern techniques of harmonic analysis (BMO and Hardy spaces, spherical maximal operator, nonlinear commutators and interpolation). Among the novelties we shall give various estimates of the Jacobian by means of the cofactors of the gradient matrix.
Functional analytic properties of subdeterminants have something to teach us about rank-one connections between Lipschitz mappings at the interface, the central issue in the geometry of microstructure and crystals. Unexpectedly, the desired rank-one connections may not be found in the polyconvex hulls of the gradients, leaving some concerns about the mathematical models of microstructure.
The nagging problem concerning Morrey's conjecture for 2x2 -matrices seems to lay beyond the power of the existing methods, but numerous links to other outstanding questions in analysis are helping drive the subject. Among them we shall discuss the fundamental L^p-inequality of the Jacobian, as a base for the analytic theory of quasiconformal mappings.
We shall briefly mention recently discovered phenomena about Jacobian of mappings between Riemannian manifolds.