in this talk i will try to prove de giorgi’s theorem on the regularity of minimal caccioppoli sets(theorem 8.4). he used a measure theoretic method closely related to properties of function of bounded variation, which is technical and powerful. i will focus on chapter 5-8 in e. giusti’s book. it is based on m. miranda’s simplification of de giorgi’s original proof.

this will be a continuation of the talk given on september 9th.

given two actions of a countable, discrete group $g$ on probabilty space $x,y$ there is a notion of when the action on $x$ is weakly contained in the action on $y$ (analogous to weak containment of representations) due to kechris: it roughly says that any finitary piece of the action of $g$ on $x$ can be approximated by some finitary piece of $g$ on $y$ (equivalent the measure on $x$ is a weak* limit of the factors of the measure on $y$). we then say that two actions are weakly equivalent when each is weakly contained in the other. we study when algebraic actions of $g$ (i.e. an action by automorphisms on a compact, metrizable, abelian group) are weakly equivalent to bernoulli shifts and find a natural class of actions related to invertible convolution operators on $g$. as part of our work, we also give conditions under which such actions are free.