36 problems

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

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 $N>1$, 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 $r>1$, 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 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 $(I_i)\in\{0,1\}^k$ 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…

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…

Let $\operatorname{Li}_n(z)$ denote the polylogarithm of positive integer order $n$, and let $i=\sqrt{-1}$. Let $\textsf{CMZV}^{2}_{n}$ denote the space of cyclotomic multiple zeta…

Pour $n\geq 3$, soit $\mathcal{A}_n$ l’associaèdre du polygone $\Pi_n$, $C_\bullet(\mathcal{A}_n)$ son complexe cellulaire, et $\mathcal{C}(F_n)$ la cogèbre de Lie motivique of the…

Pour un corps $F$, soit $D_k\mathcal{C}(F)$ la filtration croissante par la profondeur, définie par $D_{-1}\mathcal{C}(F)=0$, $D_0\mathcal{C}(F)=\mathcal{C}_1(F)$ et la condition s…

Pour un corps $F$, notons $\mathcal{C}(F)$ la cogèbre de Lie motivique et $\operatorname{Li}^{\mathcal{C}}_{n_1,\ldots,n_r}(x_1,\ldots,x_r)$ les polylogarithmes multiples motivique…

Let $N$ be a positive integer and let … For a word $w\in\{x_0,x_1\}^\ast$ of weight $n$, let $\operatorname{Li}_w(x)$ denote the corresponding multiple polylogarithm, and let…

For $k,d\in\mathbb{Z}_{\geq 0}$, define … … Let $\mathcal{C}_k$ and $\mathcal{G}_k$ be the spaces spanned by multiple Clausen and multiple Glaisher values of weight $k$, respective…

Consider convergent sums of multiple harmonic values that are both $p$-adically and weight-adically convergent. Generation conjecture for multiple harmonic-value relations. Every r…

Goncharov's conjecture. There are rational functions $f_j(x,y,z)$ such that

Let $G$ be a primitive log-divergent $\phi^4$ graph with loop number $h_G$ and $c_2(G)=-z_3$, and let $\delta_m$ be the derivation with respect to the modified parity basis in the…

Let $G$ be a primitive log-divergent graph with loop number $h_G$ and $c_2(G)=-z_3$. $c_2=-z_3$ conjecture. … The three known examples lie in the indicated space, and the source ob…

Let $G$ be a primitive log-divergent $\phi^4$ graph with loop number $h_G$ and completion $\overline{G}$, and suppose $P(G)\in\mathcal{Z}_N[\xi_N]$ for some $N$. Let…

Generalized parity conjecture. If $\left\lvert\boldsymbol{n}\right\rvert+d$ is odd, then the real part $\operatorname{Re}\operatorname{Li}_{\boldsymbol{n}}(\boldsymbol{z})$ has dep…