28 problems

Let … be the meta-profinite fundamental group of the Teichmüller tower, and let … be its Grothendieck–Teichmüller automorphism group. The absolute Galois group is denoted by…

Let $\mathfrak{r}^{\mathrm{ell}}$ be the elliptic Grothendieck–Teichmüller Lie algebra, let $\mathfrak{sl}_2$ be the degree-zero symplectic Lie algebra, and let $\delta_{2n}$,…

Let the Teichmüller tower be the collection of mapping class groups of surfaces of variable finite topological type, together with the natural homomorphisms between them induced by…

Let $P$ be the algebra of periods and let $I_{\epsilon_1,\dots,\epsilon_n}$ be the integrals appearing in Drinfeld's associator. Consider the subalgebra of $P$ generated by…

Let ${\mathfrak{grt}}$ be the Lie algebra of series in the free formal $\mathbb Q$-Lie algebra ${\mathfrak{fr}}$ satisfying the antisymmetry, cocycle, and compatibility relations d…

Grothendieck–Teichmüller–double-shuffle conjecture. The evidence suggests an isomorphism

Let the Teichmüller tower be the collection of fundamental groups $\pi_1(\mathcal{M}_{g,n})$ of the moduli stacks, together with their natural maps. Grothendieck's conjecture. The…

Let $\mathsf{GC}^{\mathrm{div}}$ be the divergence-free subalgebra of Kontsevich's graph-complex Lie algebra, and let $\mathsf{GC}_{\mathrm{(g)}}$ be the surface graph complex. The…

Let $Z_{(g)}$ and $B_{(g)}$ be the cocycles and coboundaries defining $\mathfrak r_{(g)}=(Z_{(g)}/B_{(g)},\\{\\,\\ \\})$. Let $V_{\alpha,\beta,\gamma}$ be the derivations from the…

Let $\mathfrak{r}_{(g)}$ denote the Grothendieck–Teichmüller Lie algebra of a surface of genus $g$, defined as $(Z_{(g)}/B_{(g)},\\{\\,\\ \\})$, and regard…

Let $g\geq 2$, let $H^*$ be the decorated symplectic space used to define the surface graph complex $\mathsf{GC}_{\mathrm{(g)}}$, and let $\mathfrak{h}'$ be the Lie subalgebra gene…

Let $\widehat{\operatorname{GT}}$ be the profinite Grothendieck–Teichmüller group, defined using the first two levels of the genus-zero stage of the profinite Grothendieck–Teichmül…

Let $\hat{\mathfrak T}^{\mathrm{out}}$ be the profinite Grothendieck–Teichmüller tower, viewed as a category with profinite groups as objects and outer continuous homomorphisms as…

The Grothendieck–Teichmüller Lie algebra is the Lie algebra $D524\mathfrak{grt}_1$; for each $j\geq 1$, let $D524\sigma_{2j+1}$ denote the specified non-trivial odd elements. Let…

Let $\overline{\mathcal{M}}_{*,*}$ be Getzler--Kapranov's modular operad, and let its profinite completion be equipped with its homotopy automorphism group. Let $\Lambda$ be the su…

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 $\mathfrak{grt}_{\rm ell}$ be the elliptic Grothendieck–Teichmüller Lie algebra, and let $\mathfrak{krv}^{(1,1)}$ be the Kashiwara–Vergne Lie algebra for a surface of genus $1$…

Willwacher's graph-complex vanishing conjecture.

Drinfeld's conjecture.

Injection conjecture. The Lie algebra $\mathfrak{grt}_{\rm ell}$ injects into $\mathfrak{krv}^{(1,1)}$. This conjecture is motivated by the preceding proposition, which gives an in…

Let $K_*=(K_n)_n$ be the tower of motivic groups arising from the moduli spaces $\mathcal{M}_{0,n}$, and let $\mathcal{R}_{\mathrm{Betti}}(K_*)$ and…

Let $D\simeq\mathcal{M}_{0,n_1}\times\mathcal{M}_{0,n_2}$ be a codimension-one boundary component, and let $K^{\bullet}_{\mathbb{Z},n_1,n_2}$ denote the geometric derived group ass…

Let $q$ be a primitive root of unity of order $l$, so that $q^l=1$. Consider the super-divergence and $q$-divergence cocycles defined on the Lie subalgebras of $\mathfrak{grt}_1$ c…

Let $A(M^{\delta}_0)$ be the cooperadic algebra associated with Brown's compactified moduli spaces, let $\mathsf{coGer}$ be the coGerstenhaber cooperad, and let…

Let $\boldsymbol{\boldsymbol{zeta}}_w\subset\mathbb{R}$ be the $\mathbb{Q}$-vector subspace spanned by multiple zeta values of weight $w$, with $\boldsymbol{\zeta}_0:=\mathbb{Q}$,…