An introduction to distributions and currents

Abstract

These notes are intended as a somewhat vague and preliminary introduction to some mathematical tools which have proven to be very useful in analysis, geometry and dynamics. The concepts explored here are due to Laurent Schwarz and Georges de Rham. The first is responsible for the concept of distribution and the latter made enormous contributions to topology and to this end he developed the concept of current, and both are in fact very much related. These are subjects which permeate several areas of Mathematics and we chose to give an idea of how this can be of use in the geometric theory of foliations. What we present here is not at all a self contained text, on the contrary, we jump from very elementary results to deep ones in the hope that the reader will grasp these ideas and their usefulness.