24 problems

Let ${\cal FC}$ be the algebra of formal cell-zeta numbers, let $\zeta_2$ be the element whose period is $\zeta(2)$, and let $\widehat{\mathcal{M}}(\mathbb Z)$ denote the completed…

Let $P$ be the algebra of periods, let $P_{\mathbb{Z},\mathrm{Tate}}$ be the subalgebra generated by $(2\pi i)^{\pm1}$ and periods of mixed Tate motives unramified over…

Let $P_{\mathbb{Z},\mathrm{Tate}}$ be the subalgebra of periods generated by $(2\pi i)^{\pm1}$ and periods of mixed Tate motives unramified over $\mathsf{Spec}(\mathbb{Z})$. Let th…

Let $F_{\varepsilon}=F[\varepsilon]/(\varepsilon^2)$, and let ${\cal Q}_{\bullet}(F_{\varepsilon})$ be the Lie coalgebra combining the non-Euclidean and Euclidean scissor-congruenc…

Let $F$ be a number field, let ${\cal O}_{F,S}$ be a ring of $S$-integers, and let ${\cal A}_{\bullet}({\cal O}_{F,S})$ be its motivic Hopf algebra. Let…

The motivic Hopf algebra ${\cal A}_{\bullet}(F)$ is built from motivic iterated integrals, and the map in the source identifies the corresponding motivic iterated-integral construc…

Let ${\cal Z}$ be the commutative algebra over $\mathbb Q$ spanned by multiple zeta values, and let ${\cal Z}_k$ denote its weight-$k$ subspace. Let ${\cal F}(3,5,...)_{\bullet}$ b…

Let $F$ be a field and let … The maps … are required to make the induced sequence $G_\bullet(F)\to\Lambda^2G_\bullet(F)\to\Lambda^3G_\bullet(F)\to\cdots$ a complex. Goncharov's mot…

For distinct prime numbers $p$ and $q$, let $H_{p,q}$ be the subgroup of $(\mathbb{Z}/p\mathbb{Z})^{\times}$ spanned by the images of $q$ and $-1$ in $\mathbb{F}_{p}^{\times}$, and…

For $N\in\mathbb{Z}_{\geq1}$, let $\kappa(N)$ be the dimension of the cokernel of the weight-two map $D_{2}^{\mathrm{iter}}$. Let $a,b\in\mathbb{Z}_{\geq0}$. Vanishing conjecture f…

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 $F$ 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 $\mathcal{MTM}(\mathbb{Z})$ be the Tannakian category of mixed Tate motives over $\mathbb{Z}$, and let $\mathcal{G}_{1}=\mathbb{G}_{m}\ltimes\mathcal{U}_{1}$ be its motivic Gal…

Let $f:X\longrightarrow C$ be a proper semi-stable degeneration over a smooth curve $C/k$, with $k\subset\mathbb C$. Let $X_t$ be a fiber at a complex point $t\neq s$, and write it…

The p-adic period conjecture. For every open integer scheme $Z$, the period map $\operatorname{per}_{\pf}\colon A(Z)\to\mathbb{Q}_p$ is injective. This is presented as a $p$-adic a…

Motivic Grothendieck–Teichmüller criterion. The inclusion

Let $G$ be a primitive log-divergent $\phi^4$ graph with loop number $h_G$ and $c_2(G)=0$. Suppose $P(G)$ is mixed Tate. Refined weight-mixing conjecture. Either $P(G)$ has pure we…

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 $Z$ be a totally real integer scheme, and let $\mathfrak{p} in Z$ be a totally split prime. For each $n$, let $X(\mathcal{O}_\mathfrak{p})_n$ denote the associated polylogarith…

Let $D(X,J)$ be the higher deformation associated with a simple normal-crossings stratum $Y_J$ and let $D^0(X,J)$ be its open deformation. Successive specialization along the…

Let $k$ be a number field. Given hyperplanes $L_i$ and $M_i$, for $i=0,\ldots,n$, in general position in $\mathbb P^n$ and defined over $k$, consider the mixed Tate motive … Its re…

The formal cell-zeta isomorphism conjecture. The map $m$ is an isomorphism. This would identify formal cell-zeta values, modulo their motivic relations, with the framed mixed Tate…

A residue of a Feynman integral is obtained by removing the divergent Gamma factor; the remaining integral may still require renormalization. A mixed Tate motive is a motive built…