55 problems

Let $r>1$, let $x_1,\ldots,x_r$ be roots of unity, let $a\in\mathbb{C}\setminus\mathbb{Z}$, and let $k_1,\ldots,k_r\geq1$ with $(k_r,x_r)\neq(1,1)$. Parity conjecture. The combinat…

Let $F$ be a field, and let $\mathcal{H}^{\mathcal{M}}(F)$ and $\mathcal{H}^{\mathrm{f}}(F)$ denote the motivic and formal Hopf algebras of multiple polylogarithms. Quotient each b…

Let $F$ be the field under consideration, let $k$ be a positive integer, and let $\mathcal{G}_k$ be the quotient of $\operatorname{gr}_k^\mathcal{D}\mathcal{L}_{2k}(F)$ by the subs…

Let $F$ be the field under consideration, let $k$ be a positive integer, and let $x_1,\ldots,x_k$ be arguments for the motivic multiple polylogarithm…

Let the classical polylogarithms be defined by … and let … be the multiple polylogarithm appearing in the claim. Duhr's conjecture. All multiple polylogarithms of weight at most…

Dimension-equality conjecture. For all weights $w$ and levels $N$,

Let ${\cal Z}(N)$ be the algebra of multiple polylogarithm values at $N$-th roots of unity, filtered by weight, and let $UC_{\bullet}(N)^{\vee}$ denote the graded dual of the unive…

Let $Li_{n_1,\ldots,n_k}(x_1,\ldots,x_k)$ be a multiple polylogarithm, and let…

The authors' conjecture.

Representation isomorphism conjecture. The corresponding representations of $\mathcal{Z}_q$ and $\widetilde{\mathcal{H}_{GT}}$ on quantum field theories on $S^{4,\theta}$ are isomo…

Let $N$ be a positive integer. Let ${\cal Z}^{\cal H}_{\bullet}(\mu_N)$ be the commutative graded Hopf algebra generated by framed Hodge-Tate structures associated with multiple po…

Let $N$ be a positive integer, let $\boldsymbol{}[N]$ denote the group of $N$-th roots of unity, let ${\rm Gr}{\cal G}_{\bullet \bullet}^{(l)}(\boldsymbol{}[N])$ be the associated…

Let $X$ be a smooth curve, let $H^1(X)$ denote its first cohomology, and let $G(a_1,\ldots,a_{m+1})$ denote the depth-$m$ multiple polylogarithms on $X$ constructed in the source.…

Let $N>1$, and consider multiple polylogarithms evaluated at $N$-th roots of unity. The double-shuffle and distribution relations are families of relations among these values. Prim…

Let $x_1,\ldots,x_r$ be roots of unity and let $a\in\mathbb{C}\setminus\mathbb{Z}$. For $\boldsymbol{k}=(k_1,\ldots,k_r)\in\mathbb{N}^r$ with $(k_1,x_1)\neq(1,1)$ and…

Let motivic multiple zeta values be the elements represented in the motivic Hopf algebra of multiple zeta values, and let actual multiple zeta values be the periods arising from mu…

Let $F$ be a field, and let $\mathcal{H}(F)$ be the Hopf algebra of multiple polylogarithms. Consider the quotient … An element is primitive if its reduced coproduct vanishes. Gonc…

Let $v_1,\dots,v_d$ be vectors in general position, let $w_1,\dots,w_d$ be vectors with dual basis of linear functionals $w^1,\dots,w^d$, and let $x\in\mathrm{T}_d$. For positive i…

For positive integers $n_1,\dots,n_d$ with weight $n=n_1+\dots+n_d$, let $\operatorname{Li}_{n_1,\dots,n_d}(x_1,\dots,x_d)$ be a multiple polylogarithm of depth $d$. Goncharov–Rude…

Let $F$ be the field under consideration, let $k$ be a positive integer, and let $(I_i)\in\{0,1\}^k$ specify which arguments are specialized to $1$. Let…

Let $F$ be the field under consideration, let $k$ be a positive integer, let $x_2,\ldots,x_k\in F^\times$, and let $w_0,\ldots,w_4\in\mathbb{P}^1(F)$. For each $i$, omit $w_i$ from…

Multiple polylogarithms satisfy functional equations, including those arising from the quadrangular polylogarithm functional equation. Functional-equation conjecture. All functiona…

Let $K$ be the field and let $\Sigma$ be the index set used to define the spaces of multiple polylogarithms and multiple zeta values. Let … be the map obtained by composing the map…

Let $\mathbb L(F)$ be the Lie coalgebra of multiple polylogarithms over a field $F$, with cobracket $\delta$, and let $\mathbb L^{\operatorname{symb}}(\mathbb C)$ be its symbolic v…

Let $\operatorname{Li}_n(z)$ denote the polylogarithm of positive integer order $n$, let … and let $\textsf{CMZV}^{5}_{n}$ denote the space of cyclotomic multiple zeta values of we…