12 problems

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…

Let $d\geqslant 1$ be the dimension of the base $\mathbb{R}^d$, let $k\geqslant 1$ be a differential order, and set … Let $W_d^k$ denote the Wronskian $N$-ary bracket on…

Let $N$ be a positive integer and let $\gamma_k$ be coefficients satisfying $1\leq k\leq N$. Let $\hat{\mathcal{H}}_N$ be the quantum Hamiltonian, let $\hat{\mathcal{C}}$ be the qu…

Let $P_k=\mathbb F_2[x_1,\ldots,x_k]$ be the polynomial algebra, let $Q P_k$ denote its quotient by the image of the positive-degree Steenrod algebra, and let … be Kameko's squarin…

Higher-order derivation conjecture. If

Intersection-derivation conjecture. The set of derivations of $A$ is the union of the derivations of $A_1$ and $A_2$, restricted to $A$. This conjecture concerns how derivations be…

Let $A\subseteq\mathbb{K}[x]$ be a finite-codimension subalgebra, let $k_\alpha$ be the dimension of the vector space of all $\alpha$-derivations of $A$, and let…

Let $A\subseteq\mathbb{K}[x]$ be a subalgebra with spectrum containing $\alpha$, let $C=\{\alpha_1,\ldots,\alpha_s\}$ be its spectrum, and let $\alpha_j\sim\alpha$ mean that…

Let $A=\mathbb{K}[p,q]$ be the subalgebra generated by $p$ and $q$, let $\chi_{p,q}(x)$ be their characteristic polynomial, and let $a(x)$ be a polynomial of degree at least two di…

Freudenburg's conjugacy conjecture. Every maximal Lie subalgebra $\mathcal{A}\subset\operatorname{LND}(\mathbb{K}[x_1,\ldots,x_n])$ is conjugate to $\mathfrak{T}$. This is the uniq…

Let $k$ be a commutative ring, let $k[t]$ be the polynomial algebra, and for $m\geq 0$ let $HS_1(k[t])_{t^m}$ denote its $t^m$-layered symmetric homology, obtained from the decompo…

Let $B\subset\mathbb R^n$, let $w(z)$ be a weight function, and let $\{u_\alpha\mid\alpha\in\mathbb N^n\}$ be a sequence of orthogonal polynomials over $B$. Let $\mathcal M$ be the…