5 problems

Let $\Gamma\curvearrowright (X,\mu)$ be a Bernoulli action of an icc group $\Gamma$ with Kazhdan's property (T). The action is W$^*$-superrigid if its isomorphism class as a group…

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

Let $G$ be a countable group with non-vanishing first $\ell^2$-cohomology. Equivalently, suppose that there is a function $f:G\to\mathbb{C}$ with $f(0)=0$, which is not in…

Let $\Gamma$ be a countable group and let $[\,\cdot\,]\colon\Gamma\to\mathbb{Z}$ be a homomorphism. Suppose that … where $M>\sum_{s\in I}f_s$, and $[s]\geqslant1$ and $f_s>0$ for e…

Let $\Gamma\curvearrowright (X,\mu)$ be a Bernoulli action of an icc group $\Gamma$ with Kazhdan's property (T). An action is W$^*$-superrigid if every free ergodic probability-mea…