Structure of null sets in the plane, and some applications to Geometric Measure Theory

In this talk I will try to describe (part of) an ongoing reasearch with D. Preiss and M. Csörnyei. The starting point is a decomposition theorem for null sets (sets of measure zero) in the plane, which can be obtained as a direct consequence of a geometric version of the Erdös-Szekeres theorem. Then I will outline some applications to problems in Geometric Measure Theory (and not only). If time permits, I will finally address the issue of extending these results to higher dimensions.