48 problems

Let $\Phi^{\mathfrak{m}}(x,y)$ be the motivic version of the modified Drinfeld associator, and let $\Phi(x,y)$ be the corresponding modified Drinfeld associator defined in the sour…

Let $K\subset\mathbb{C}$ be a number field. The ring $\mathcal{H}(K)$ is the ring of effective motivic periods of the category of mixed Tate motives over $K$, and motivic iterated…

Let $F$ be a number field. For elements $a_0,\ldots,a_{n+1}\in F$, let … be the motivic iterated integral associated with the corresponding motivic fundamental groupoid, and let…

Let $\mathcal{MT}(\mathbb Z)$ be the category of mixed Tate motives over the integers, let $G^{\mathrm{mot}}_{\mathcal{MT}(\mathbb Z)}$ be its motivic Galois group, and let…

Panzer–Schnetz coaction conjecture. The Galois coaction closes on $\phi^4$-periods:

Let $\mathcal{M}$ be the fixed category of motives, with de Rham and Betti fibre functors, motivic period algebra $\mathcal{P}^{\mathfrak{m}}$, and complex period map … A complex p…

Let $a_i,c_i\in\mathbb{N}^{\ast}$ with $c_i\neq 2$, and define … where $\delta_c=\delta_{c=1}$ equals $1$ if $c=1$ and $0$ otherwise. Define … … and … Here \text{\boldmath 1…

Let ${\mathcal Z}$ be the algebra over $\mathbb{Q}$ spanned by multiple zeta-values, graded by their weight. Let ${\mathcal F}(3,5,\dotsc)_\bullet$ be the free graded Lie algebra g…

Let $M(\bs;\varepsilon)$ denote a multiple mixed value and $T(\bs)$ a multiple t-value. Classify indices as wide, slender, or narrow according to whether their largest component is…

Let $M(\bs;\varepsilon)$ denote a multiple mixed value and $T(\bs)$ a multiple t-value. Classify indices as wide, slender, or narrow according to whether their largest component is…

Let $M(\bs;\varepsilon)$ denote a multiple mixed value of depth $4$, let $T(\bs)$ denote a multiple t-value, and classify indices as wide, slender, or narrow by whether their large…

Let $\bfs$ be an index, let $T(\bfs)$ be a multiple t-value (MTV), and let $\bfs^*$ be its dual. Define the width $\operatorname{wth}(\bfs)$ as the largest component of $\bfs$; cal…

Let $\bfs$ be an index and let $\bfs^*$ denote its dual. Let $T(\bfs)$ be a multiple t-value (MTV), and call it unramified when it belongs to the unramified class considered in the…

Let $d$ be a positive integer with $d\ge 3$. Write $1_{d-3}$ for a string of $d-3$ entries equal to $1$, let $\zeta^\sharp$ denote the Euler sharp sum, and let $\zeta(m,n)$ denote…

Maximal transcendence-degree conjecture. The transcendence degree of the $\mathbb Q$-algebra generated by this set is $d(a,b)$.

Maximal transcendence-degree conjecture. The transcendence degree of the $\mathbb Q$-extension field generated by this set is maximal.

The period conjecture. The period map is injective.

Yamashita's conjecture. This $\mathbf{Q}$-algebra is free on the obvious generators $\zeta_p(3),\zeta_p(5),\zeta_p(7),\ldots$. The conjecture is a $p$-adic analogue of the expected…

Grothendieck's period conjecture. This period map is injective. A consequence is that the $\mathbf{Q}$-algebra generated by $\zeta(2)$ and the odd values $\zeta(2k+1)$ for $k\geq1$…

Unramified MtV conjecture. For $m,n,e\text{\in}N$ and with all components of $q$ at least two, the MtV

Kaneko–Tsumura's conjecture. For even weights, the values $T(k,l,m)$ with $k,r$ odd and at least $3$ and $q$ even, together with their duals, are in $MZV$. Together with the single…

Let $a\in\mathbb{N}_0$, $\ell\in\mathbb{N}$, and let $u_j,v_j\in X:=\{1,\bar2,3,\bar4,\dots\}$ for $j=1,\dots,\ell$, with … For $\mathfrak S_\ell$ the symmetric group on $\ell$ let…

Brown's subdivided-edge invariance conjecture. This morphism factors through the quotient by the relation $G\sim G'$ whenever $G'$ is obtained from $G$ by subdividing an edge.

Let $\mathcal{H}_4$ be the algebra of level-$4$ motivic multiple zeta values, and let … be its period map. Period-map injectivity conjecture. The period map…

Let $p_{a,b,c}$ and $q_{a,b,c}$ be the polynomials and let $r_{a,b,c}$ and $r'_{a,b,c}$ be the integers defined in the statement. Let $I_{\mathrm{bl}}^{\mathfrak{m}}$ denote the mo…