11 problems

Let ${\cal A}(P)$ be the algebra containing the elements $P_m$, for $m\in\mathbb{N}$, and let $\Phi$ be the map from ${\cal A}(P)$ to expressions for the noncommutative KP hierarch…

Dimension conjecture. The correct dimension is $\dim X$ in both cases; that is,

Let $M$ be a symplectic $C^\infty$-manifold or an affine complex algebraic variety, let $G$ be a finite group acting on $M$ by symplectic transformations, and let $B$ be a $G$-equi…

Let $X$ be either $A_{n-1}$ or $D_n$, and let $\theta_1^X,\ldots,\theta_n^X$ be the corresponding elements of the quadratic algebra $\Lambda_{quad}(X)$. Write $\epsilon_k$ for the…

Let $n$ be odd, let $\tilde{d4ad4e}_n$ be the left ideal described in the preceding corollary, and let $\tilde{M}_i$ denote its elements indexed by $0d4a i\frac{n-1}{2}$. Dimension…

Let $B$ be a non-commutative algebra generated by $n$ degree-one elements with $r$ homogeneous quadratic relations; this is an algebra of type $(n,r)$. For a power series…

Let CLWE denote learning with errors based on cyclic algebras, and let RLWE denote ring learning with errors. The CLWE construction uses the non-commutative multiplication of a cyc…

Let $C$ be a smooth curve of genus one embedded in ${\mathbb P}^{n-1}$ by the complete linear system of a $K$-rational divisor of degree $n\geq 3$. Let $A$ be the associative $K$-a…

Let $G < \operatorname{GL}(2, \mathbb{C})$ be a nonabelian finite subgroup and let $R=\mathbb{C}\llbracket x,y\rrbracket^G$. A finite-dimensional local $\mathbb{C}$-algebra is an a…

Bispectral algebra conjecture. The algebra of all such $\Theta(x)$ is the algebra of polynomials

Tikaradze's analogue of the Kac–Weisfeiler conjecture. Suppose that $A$ is a nonnegatively filtered $\bold{k}$-algebra such that $\operatorname{gr} A$ is a finitely generated commu…