7 problems

Popa's conjecture. The von Neumann algebra $L^\infty(X^\Gamma)\rtimes\Gamma$ has a unique Cartan subalgebra, up to unitary conjugacy.

Let $\mathbb F_n$ act freely and ergodically by probability-measure-preserving transformations on $(X,\mu)$, where $n\geq 2$. The associated group measure space factor is…

Let $\mathbb{F}_n \curvearrowright (X,\mu)$ be an action of a free group on a probability space, with no profiniteness assumption. The crossed product…

Let $G$ be a groupoid, let $G'$ denote its isotropy subgroupoid, and let $S$ be an open subgroupoid of $G'$ such that the associated twist restriction $\mathcal{E}_S$ is abelian. P…

Let $\mathcal E$ denote the class of existentially closed II$_1$ factors. A Cartan subalgebra is a maximal abelian subalgebra whose normalizer generates the factor. Cartan-free con…

Let $\Gamma$ be a countable group with positive first $\ell^2$-Betti number, so that … Let $\Gamma\curvearrowright (X,\mu)$ be a free ergodic probability-measure-preserving action,…

Premet's conjecture. There exists a nonempty Zariski open subset $V\subseteq{\mathfrak g}$ consisting of regular elements such that, for any $u,v\in V$, the Cartan subalgebras