21 problems

Let multiple zeta values be the real numbers … for $d\in\mathbb{N}^{\ast}$ and $s_d,\ldots,s_1\in\mathbb{N}^{\ast}$ with $s_d\geq 2$. The associator equations are the polynomial eq…

Let $L=(G,\circ)$ and $M=(G,*)$ be Moufang loops on the same set $G$ of order $n$, and define their distance by … Suppose that $L$ and $M$ have nonequivalent associators. Distance-…

Racinet's conjecture. For $N=1$, there is a morphism of torsors

Alekseev–Torossian's conjecture. They conjecture that

Let $C$ be a smooth curve, let $\operatorname{Conf}(C,n)$ be the configuration space of $n$ distinct points on $C$, and let $\operatorname{Conf}^f(C,n)$ denote its framed version.…

Let $\mathcal M^{B}_{un}(Ran(\mathbb A^1))$ and $\mathcal M^{dR}_{un}(Ran(\mathbb A^1))$ be the Betti and de Rham categories equipped with their natural tensor structures and fiber…

Product decomposition conjecture. The characteristic function of any multiplicity-free cnu contractive associator in $B_n^*(\mathbb D)$ is the pointwise product of the characterist…

The polyzeta values are the coefficients of the Knizhnik--Zamolodchikov associator, a group-like power series, and they satisfy equations called the regularized double shuffle rela…

Let $\mathfrak{p}(U):=\mathbb L(\mathsf{x}_0,\mathsf{x}_1)^{\wedge}$ be the lower central series completion of the free Lie algebra in variables $\mathsf{x}_0,\mathsf{x}_1$, and le…

Let multiple zeta values be the real numbers … for $d\in\mathbb{N}^{\ast}$ and $s_d,\ldots,s_1\in\mathbb{N}^{\ast}$ with $s_d\geq 2$. The double shuffle relations are the regulariz…

Multiple-zeta-value quotient conjecture. There is an isomorphism

Knizhnik–Zamolodchikov associator conjecture. The universal $A_{\infty}$ structure constructed in the paper corresponds to choosing the Knizhnik–Zamolodchikov associator.

Let $Phi$ be Drinfeld's associator, with formal variables $e_0,e_1,e_{\infty}$ satisfying $e_0+e_1+e_{\infty}=0$, and let Drinfeld's 2-cycle, 3-cycle (hexagon), and 5-cycle (pentag…

Let $A(M^{\delta}_0)$ be the cooperadic algebra of Brown's compactified moduli spaces, let $\boldsymbol{\mathcal{Z}}_{\scriptstyle{\mathrm{ns}}}(A(M^{\delta}_0))$ be the associated…

Let $A(M^{\delta}_0)$ be the cooperadic algebra of Brown's compactified moduli spaces, let $\boldsymbol{\mathcal{Z}}_{\scriptstyle{\mathrm{ns}}}(A(M^{\delta}_0))$ be the associated…

Generalized Kashiwara–Vergne conjecture. There exists a group automorphism $P:\exp\mathfrak F_2\to\exp\mathfrak F_2$ such that $P\in T\operatorname{Aut}\exp\mathfrak F_2$,

Let $M$ be the scheme of associators, let ${\bf k}_{MZV}$ be the ${\mathbb Q}$-ring associated with multiple zeta values, and let $\Phi_{KZ}\in M({\bf k}_{MZV})$ be the KZ associat…

Transcendence Conjecture. The morphism $\varphi_{KZ}$ is injective. This asserts algebraic independence of the coordinate functions of the associator in the sense encoded by the co…

Let $X=\{x_0,x_1\}$, let $A$ be a ${\mathbb Q}$-extension, and let $\Phi$ be a noncommutative formal power series on $X$ with coefficients in $A$, subject to suitable conditions. L…

Let $\overline{C}_{n,0}$ be the compactified configuration space of $n$ points in the upper half-plane, and let $\omega_n^\infty$ be the restriction of the connection $\omega_n$ to…

Let $GRT_1$ be the Grothendieck–Teichmüller group and let $DMR_0$ be the double shuffle group. Let $G^{\mathrm{mot}}_{\mathbb{Z}}$ denote the motivic Galois group of $\mathbb{Z}$,…