41 problems

Slope subalgebra conjecture. The slope subalgebra $\mathcal{B}_{\mathbf{m}}$ is isomorphic to the quantum group

Let MZVs be the convergent multiple zeta values, with relations generated by products of their iterated-integral and nested-sum descriptions. The resulting identities are called sh…

Shuffle-product completeness conjecture. Under the shuffle product relations, Corollary 5.10 for $m=1,2$ gives all $\mathbb{Q}$-linear relations among multiple zeta values.

Let $\mathfrak H=\mathbf Q\langle x,y\rangle$, with shuffle multiplication (denoted by juxtaposition with the shuffle symbol) and harmonic multiplication $*$. Let $\mathfrak H^0$ b…

Let $T(mathbb R^d)$ be the tensor algebra equipped with the left and right halfshuffle products, and let $\mathcal{I}(\mathbf{PL}^d_{\leq 2})$ denote the ideal associated with piec…

Nekrasov's localized wheel-condition conjecture. The inclusion of the tripled-quiver CoHA image into the wheel-condition algebra becomes an equality after localization:

Nekrasov's wheel-condition generation conjecture. The localized wheel-condition algebra equals the localized spherical subalgebra:

Let $\mathfrak{H}^0$ be the admissible word space, let $\mathfrak{H}^0_\ast$ denote its harmonic-product algebra, and define…

The derivative and relations conjecture for multiple Eisenstein series. For every $w\in\mathfrak{H}^{\geq2}$,

Let $M_{n,k}^{\mathrm{prim}}$ be the primitive quotient and let the preceding primitive vector-space conjecture identify it with $H^*(X)[\sum_{i=1}^nx_i,\sum_{i=1}^ny_i]$. Ring-str…

Generators conjecture. The space $M_{n,k}^{\mathrm{prim}}$ is spanned by products of $1/\Delta_x^{k-1}$ and polynomials of the forms $\psi_s(\mathrm{pt})$ and $\psi_s(1)$. The clai…

Let $M_{n,k}=H_L^n(O(k))^\vee$, and let $M_{n,k}^{\mathrm{shuffles}}\subset M_{n,k}$ be spanned by shuffle products $f*g$ with $f\in M_{n',k}$, $g\in M_{n-n',k}$, and $0<n'<n$. Def…

Let $\mathbb{Q}\subseteq\mathcal{C}$ and let $\mathcal{R}_\mathrm{T}$ be freely generated by $\{a_i\mid i\in I\}\subseteq\mathcal{R}_\mathrm{T}$ as a $\mathcal{C}$-module. Let…

Let $\mathbb{Q}\subseteq\mathcal{C}$, let $\mathcal{R}_\mathrm{T}$ be generated as a $\mathcal{C}$-module by $\{a_i\mid i\in I\}\subseteq\mathcal{R}_\mathrm{T}$, and let $\le$ be a…

Quantum toroidal superalgebra isomorphism. There is an isomorphism

Kawashima's completeness conjecture. The family of linear relations obtained by expanding the product on the left-hand side of Kawashima's identity using the shuffle product formul…

Let $\mathcal{Z}^f$ be the algebra of formal multiple zeta values, defined as a quotient of the shuffle algebra by the ideal generated by the extended double-shuffle relations, and…

Spherical shuffle realization conjecture. The homomorphism $\xi$ can be promoted to an algebra isomorphism

Shuffle-of-areas closure conjecture. If all $i_j$ are distinct, then

Let $\mathsf{ConjInv}$ be the algebra of conjugation invariants, and call a homogeneous element a generator at level $n$ if it is one of the homogeneous generators used for this sh…

Let $T(\mathbb{R}^d)$ be the tensor algebra, let $S$ denote the shuffle algebra of shuffle products of letters, and let $\mathsf{ConjInv}$ denote the conjugation invariants. Define…

Let $Q$ be an orientation of a type ADE Dynkin diagram, and let a total convex order on $\Delta^+$ refine the Auslander–Reiten partial order. Let $f_\alpha(x)$ be the fused current…

Let $U_q(L\mathfrak{g})$ be the quantum loop group of a finite-type ADE Lie algebra, with negative half $U_q(L\mathfrak{n}^-)$ and a completion $\widehat{U}_q(L\mathfrak{n}^-)$, an…

Shuffle-algebra image conjecture.

Hecke deformation conjecture. Corollary 28 and Corollary 29 both seem to hold in $\mathcal{H}$ when every $t_\ell$ is replaced by $t_\ell^{\mathcal{H}}$.