23 problems

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…

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

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

Ihara–Kaneko–Zagier conjecture. All algebraic relations in the algebra $\mathcal{Z}$ are obtained from the comparison of the extended maps

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…

Quantum toroidal superalgebra isomorphism. There is an 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 $U_q(L\mathfrak{g})$ 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 $t_\ell^{\mathcal{H}}$.

Let $\mathfrak{X}_{\mathrm{ev}}'$, $\mathfrak{X}_{\mathrm{ev}}^+$, $\mathfrak{X}_{\mathrm{od}}$, $\mathcal{H}$, and $L_B^\circ$ be the parity-graded spaces and block-regularization…

Let $\mathcal A$ be the algebra defined by the action-product map from the shuffle-algebra construction, and let $U_q(\widehat{\mathfrak g}^Q)$ be the Hopf algebra obtained from th…

Let $Q$ be a quiver, let ${\mathcal B}_{\b0}$ be the slope-zero algebra graded by dimension vectors, and write … where $A_{Q,\boldsymbol{n}}(t)$ is the Kac polynomial. Define … and…

Let $\Delta^+$ be the set of positive roots, let $d\in\mathbb{Z}$, and let $\ell(\alpha,d)$ denote the Lyndon loop word of degree $(\alpha,d)$. Let $\Phi^L$ be the homomorphism fro…

Let $A$ be a primitive bialgebra in a braided monoidal abelian category, and let ${\mathbb{B}\mathbb{B}}_n(A)$ denote its $n$th bibar-cube, viewed as a double representation of…

Let $(n,k)$ and $(n',k')$ be lattice points, and let $e_{n,k}$ and $e_{n',k'}$ be the corresponding geometric operators. Let $\Delta_*$ denote pushforward along the diagonal, let…

Let $S$ be the surface under consideration, let $\CV_\emph{sm}$ be its small shuffle algebra, and let $K_\CM$ denote the entire K-theory group in place of the tautological part…

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…