6 problems

Let $R$ be an algebra over a field $K$, let $z$ be a free generator, and let $x_1,\dots,x_n$ be noncommuting free generators. Write $K\langle x_1,\dots,x_n\rangle$ for the free ass…

Let $R$ be an algebra over a field $K$, let $z$ be an indeterminate, and let $x_1,\dots,x_n$ be commuting indeterminates. Zariski's cancellation conjecture. If … then … The conject…

Makar-Limanov invariant stability conjecture.

Cancellation conjecture.

Let $R$ be a $\mathbb{K}$-algebra, and let $\mathbb{K}\langle x_1,\ldots,x_n\rangle$ and $\mathbb{K}\langle y\rangle$ denote free associative algebras. Free associative cancellatio…

Let $X$ be a smooth affine surface. Say that $X$ has negative logarithmic Kodaira dimension when its logarithmic Kodaira dimension is negative. Say that $X$ fails the…