7 problems

Fundamental-group conjecture. The only uncountable subgroup of $\mathbb R_+^*$ that can occur as the fundamental group of a countably generated II$_1$ factor is $\mathbb R_+^*$.

Let $M$ be a separable II$_1$ factor, and let $P\subset M$ be a regular von Neumann subalgebra, meaning that the normalizer of $P$ in $M$ generates $M$. Margulis-type conjecture. T…

A nonamenable II₁ factor is a nonamenable finite factor with separable predual. For II₁ factors $N$ and $M$, write $N\hookrightarrow M$ when there is a unital embedding of $N$ into…

Ioana's conjecture. There exist II₁ factors $M$ such that every stable self-embedding $M\hookrightarrow M^d$ is trivial. This conjecture was proven in 2013 by Deprez through an ext…

Let $M$ be a $\rm II_1$ factor with the stable single generation (SSG) property, meaning that some sequence $t_n\searrow0$ has each amplification $M^{t_n}$ singly generated. Let…

Let $M$ be a $\rm II_1$ factor such that any amplification of $M$ can be generated by two selfadjoint elements. A pair of hyperfinite factors $R_0,R_1\subset M$ is to be constructe…

Assume $\mathrm{OCA}_\infty+\mathrm{MA}_{\aleph_1}$. Let $(M_i)$ and $(N_i)$ be sequences of separably representable type II$_1$ factors, and let $\bigoplus_\tau M_i$ be the ideal…