William Allard : Currents in metric spaces (Apr 27, 2010 4:25 PM)

Motivated by the need to formulate and solve Plateau type problems in higher dimensions and codimensions, normal and integral currents were introduced by Federer and Fleming around 1960; their work was, to some extent a generalization of earlier work by DeGeorgi in codimension one as well as the work of Reifenberg in arbitrary codimensions. Since then a great deal of work has been done on the Plateau problem and related variational problems. This work has always been based on geometric measure theory. The so-called closure theorem for integral currents and the boundary rectifiability theorem are essential ingredients in all of this work; these theorems depend on the Besicovitch-Federer structure theory for set of finite Hausdorff measure in Euclidean space. More recently, in the work of Ambrosio and others, a useful theory of Sobolev spaces for functions with values in an arbitrary metric space has been developed and applied to a variety of problems. Ambrosio and Kirchheim have developed a theory of currents in metric spaces in which they are able to give geometrically appealing proofs of generalizations of the aforementioned closure and rectifiability theorems using some ideas of Almgren and DeGiorgi and avoiding the use of the Besicovitch-Federer structure theory. In this talk I will describe how they do it.

• Category: Geometry and Topology
• Duration: 01:34:52
• Date:  April 27, 2010 at 4:25 PM
• Views: 104
• Tags:   seminar, Geometry/topology Seminar