35 problems

Let $T\mathcal M(n)$ denote the additive part of the degree-$n$ component of the indecomposable space, and let $Z_n(a)$ be a codimension-$n$ cycle on … For $a\in\mathbb A^1-\{0\}$,…

Let $\widehat G_{n-1}^n$ be the configuration space used to define the Grassmannian $n$-logarithm, and let $a_i,b_j$ be the maps on this space appearing in its functional equations…

Let $F$ be a field. Let $B_2(F)$ be the Bloch group, let $R_3(F)$ be the subgroup generated by the stated relations in ${\mathbb Z}[{\mathbb P}^1_F]$, and define … Let…

Let $G_4(F)$ be the weight-four Grassmannian group, let $\mathcal B_4(F)$ be the corresponding polylogarithmic group, and let … be the canonical cobracket described in the construc…

Let $\widetilde A_n$ be the free abelian group generated by admissible pairs of simplices, let $\overline C_{2n}$ be the Grassmannian configuration group, and suppose the maps…

Let $F$ be a field, let $B_n(F)$ be Goncharov's polylogarithmic group, and let … be the associated map. Write $K^{(n)}_{2n-1}(F)_\mathbb Q$ for the $n$-th graded quotient of the…

Let $R_1^*$ denote the Flint Hills series and let $\tau=\frac{1-\frac{1}{2} u}{?}$ be... There exist rational constants $c1,c2 obreaktext{ in } obreakbb Q$ such that … By Theorem, thi…

Let $k$ be as in the preceding theorem, let $calpha,cbeta$ be the parameters in the narrow ideal-class partial zeta value $czeta(k,(calpha,cbeta),cscrB)$, and let $cscrB^*$ denote…

Let $F$ be an infinite field. Let $B_3(F)$ be Goncharov's Bloch group defined using the stated relations, and let $\mathcal{L}^{\mathrm{f}}_3(F)$ be the weight-three component of t…

Let $F$ be an infinite field. The truncated cobracket … is obtained by omitting the $\mathcal{L}_1(F)\wedge\mathcal{L}_{n-1}(F)$ component of the cobracket. Goncharov's classical p…

Let $F$ be an infinite field, let $1\leq i\leq n$, and let $\mathcal{H}^{\mathrm{f}}(F)$ denote the formal Hopf algebra with weight-$n$ cohomology…

Let $F$ be a number field. For $x\in F\setminus\{0,1\}$ and $n\geq 1$, let $\operatorname{Li}_n^{\mathcal{C}}(x)\in\mathcal{C}_n(F)$ be the motivic polylogarithm, where…

Let $k\geq 2$, let $a_1,\dots,a_k$ be elements for which the displayed multiple polylogarithms are defined, and let $\mathcal{L}_{2k}$ be the weight-$2k$ component of the Lie coalg…

Plectic polylogarithm specialization conjecture. The specialization satisfies

Pour $n\geq 1$, soit $\iota^\bullet(n):\mathcal{B}^\bullet(F,n)\rightarrow\mathrm{CE}^\bullet(\mathcal{C}(F))_n$ le morphisme de complexes induced by motivic polylogarithms. Conjec…

Soit $\psi_n:\ker(\delta_n)\rightarrow\ker(\delta:\mathcal{C}_n(F)\rightarrow(\Lambda^2\mathcal{C}(F))_n)$ le morphisme construit à partir des polylogarithmes motiviques, où $F$ es…

Soient $F$ un corps de nombres et $n\geq 2$. Notons $d_n=r_1+r_2$ si $n$ est impair et $d_n=r_2$ si $n$ est pair, et soit…

Finiteness conjecture. For every positive integer $N$, the number of $N$-admissible rational functions is finite. The conjecture is stated as plausible without a rigorous proof in…

Weight-five depth-one conjecture. (i) There exists a formal linear combination $\mathcal{Q}_5(v_1,\dots,v_{10})$ of rational functions on $\operatorname{Conf}_{10}(5)$ such that…

Let $F$ be a field and let $R_2(F)$ be the weight-two relation subgroup in the modified definition using $F\setminus\{0,1\}$. An inverted five term relation is a relation obtained…

Let $F$ be a field and let $R_2(F)$ be the subgroup of the relevant free abelian group generated by functional relations in weight two. For $x,y\in P^1_F$, write $FT[x,y]$ for the…

Let $K$ be the field appearing in the motivic category, and for each integer $n\geq 2$ write … For $t\in K$, let $\operatorname{Li}_n^{U,sv}(t)$ denote the single-valued unipotent…

Let … denote the third logarithmic Mahler measure of the polynomial $1+x+y$, and let $\operatorname{Ls}_4$ and $\operatorname{Cl}_s$ denote the log-sine and Clausen functions, resp…

Zagier's K-theoretic conjecture. There is a homomorphism $B_m(F)\longrightarrow K_{2m-1}(F)$ with finite kernel and cokernel such that the composition with $R_m(F)$ equals the map…

Goncharov's conjecture. The element $B(x,y;z)$ belongs to ${\cal P}_4(E)$; equivalently, it is a linear combination of $4$-logarithms. The construction gives an element killed by…