Borel reductions of profinite actions of SL(n,Z)

Greg Hjorth and Simon Thomas proved that the classification problem for torsion-free abelian groups of finite rank \emph{strictly increases} in complexity with the rank. Subsequently, Thomas proved that the complexity of the classification problems for $p$-local torsion-free abelian groups of fixed rank $n$ are \emph{pairwise incomparable} as $p$ varies. We prove that if $3\leq m<n$ and $p,q$ are distinct primes, then the complexity of the classification problem for $p$-local torsion-free abelian groups of rank $m$ is again incomparable with that for $q$-local torsion-free abelian groups of rank $n$

On the measure and the structure of the free boundary of the lower dimensional obstacle problem

We provide a thorough description of the free boundary for the lower dimensional obstacle problem in $\mathbb{R}^{n+1}$ up to sets of null $\mathcal{H}^{n-1}$ measure. In particular, we prove (i) local finiteness of the $(n-1)$-dimensional Hausdorff measure of the free boundary, (ii) $\mathcal{H}^{n-1}$-rectifiability of the free boundary, (iii) classification of the frequencies up to a set of dimension at most (n-2) and classification of the blow-ups at $\mathcal{H}^{n-1}$ almost every free boundary point

Factoriality, type classification and fullness for free product von Neumann algebras

We give a complete answer to the questions of factoriality, type classification and fullness for arbitrary free product von Neumann algebras.Comment: 20 pages; to appear in Adv. Mat