11 problems

Let $\mathbf{k}$ be an algebraically closed field, and let $\operatorname{Aut}_{\mathbf{k}}(\mathbb{A}^n)$ denote the group of automorphisms of affine $n$-space over $\mathbf{k}$.…

Conjugacy-to-triangularity conjecture. For every $\phi\in\mathcal{E}_3(k)\cap\operatorname{T}_3(k)$, there exists $\sigma\in\operatorname{T}_3(k)$ such that

Stable Tameness Conjecture. Every $\phi\in\operatorname{Aut}_k k[\boldsymbol{x}]$ is stably tame. This is a central stabilization question for polynomial automorphisms; the source…

Rips's conjecture. Linear automorphisms, together with $\psi$, generate the entire tame automorphism group of $K[x_1,\ldots,x_n]$. The characteristic-zero analogue is known, while…

Let $\mathbf{k}[x_1,\ldots,x_n]$ be a polynomial algebra and $\mathbf{k}\langle x_1,\ldots,x_n\rangle$ a free associative algebra. An automorphism is stably tame if, after adjoinin…

Let $Q$ be the affine quadric threefold and let $G$ be its tame automorphism group. Let $\mu$ be a symmetric atomic measure on $G$, let $\nu_n=\mu^{*n}$ be its $n$-fold convolution…

Let $\Bbbk$ be a field and let $\Bbbk[x_1,\dotsc,x_n]$ be the polynomial ring in $n$ variables. An elementary automorphism sends one variable $x_i$ to $\alpha x_i+f$, where…

Furter–Rogalski's conjecture. The tame automorphism group $\operatorname{TA}_n(K)$ is the amalgamated product of the subgroups $\widetilde H_1,\ldots,\widetilde H_n$ along their pa…

Let ${\bf k}[x_1,\dots,x_n]$ be a polynomial algebra and ${\bf k}\langle x_1,\dots,x_n\rangle$ the free associative algebra. An automorphism is stably tame if, after adjoining fini…

Let $k$ be a field of characteristic $0$, and let $3\leq p_1\leq d_2\leq d_3$ be integers with $p_1$ a prime number. Write $\mathrm{mdeg} F$ for the multidegree of a polynomial map…

Degree-two tameness conjecture. If