7 problems

Quantum Gel'fand–Kirillov conjecture. The skew field $\operatorname{Fract} A$ is isomorphic to the skew field of fractions of a quantum affine space over $K$.

Let $R$ be a skew field and let $G$ be a locally indicable group. When it exists, let ${\mathcal{D}}_{RG}$ denote the Hughes-free skew field of $RG$. Hughes-free skew-field existen…

Classification conjecture. Every infinite skew-field arising in a finite-dimensional theory is either commutative and real closed, commutative and algebraically closed, or quaterni…

Let $\mathbb{F}$ be a field and let $G$ be a torsion-free group. An epic skew-field embedding is an embedding whose target is generated by the image of $\mathbb{F}G$ in the relevan…

Non-amenability conjecture. The free skew field on $r$ generators over $K$ is non-amenable for every base field $K$.

Let $\nu:F^*\rightarrow\mathbb{Z}^n$ be a maximal valuation, and let $\mathcal{O}=\{f\in\mathscr{F}:\nu(f)\geq 0\}$ with maximal ideal $\mathcal{m}=\{f\in\mathscr{F}:\nu(f)>0\}$ in…

GA conjecture. If $D$ is finitely generated as a skew field over $Z$ and $D$ is infinite dimensional over $Z$, then $D$ contains a noncommutative free group $Z$-algebra.