title: On maps whose distributional Jacobian is a measure.

invited lecture at the "Summer Symposium in Real Analysis XXXI".
Trinity College, Oxford, August 12-16, 2007.

published in: Real Anal. Exchange 2007, 31st Summer Symposium 
Conference, pp. 153-162.

abstract: When the dimensions of domain and co-domain are the same, 
the Jacobian of a map is the determinant of its gradient. It was 
noticed long time ago that this nonlinear operator can be extended 
in a natural way to maps in certain Sobolev classes. 
This extension, known as distributional Jacobian, does not always 
agree with the Jacobian according to the standard (pointwise) 
definition, in the same way the distributional derivative of a 
function with bounded variation does not agree with the classical 
one. 
In particular, for map that take values in a given hypersurface, the 
pointwise Jacobian must vanish while the distributional one may not. 
If this is the case, the latter has an interesting interpretation in 
term of the (topological) singularity of the map. 
In this lecture I will review some of the basic results about maps 
whose distributional Jacobian is a measure, focusing in particular 
on maps with values in spheres. 

keywords: distributional determinant, distributional Jacobian, 
topological singularities, Sobolev maps, integral currents, 
finite perimeter sets, rectifiability.

MSC (2000): 49Q15 (53C65, 58A25)