62 problems

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…

Kontsevich's conjecture. Every graph hypersurface $\hat X_\Gamma$ should be a mixed Tate motive.

Mixed Tate motivic Galois group conjecture. The groupscheme $\mathrm{Gal}_\mathrm{mot}$ is conjectured to be isomorphic to the motivic Galois group of the Tannakian category of mix…

Let ${\rm Cor}_{\cal M}$ be a motivic correlator defined over a number field $F$, and suppose that in the situation described in the source the correspondence principle holds and t…

Let $F$ be a field. Beilinson–Deligne conjecture. For every field $F$, there should exist an abelian category of mixed Tate motives. This conjecture would provide the expected cate…

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…

Let ${\cal FZ}$ be the formal multiple-zeta algebra, defined as the quotient of the free $\mathbb{Q}$-algebra on formal iterated-integral symbols by the regularised double-shuffle…

The mixed Tate motives conjecture. The periods of all mixed Tate motives over $\mathbb{Z}$ should be expressible in terms of multiple zeta values. This is a broad motivic-period co…

Goncharov's quadric conjecture. The map $\phi^q$ is an isomorphism.

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…

A mixed Tate motive unramified over $\mathbb Z$ is a mixed Tate motive whose arithmetic ramification is absent over the integers. Its periods are complex numbers obtained by integr…

Let ${\cal H}_T(\mathbb{Q}_{\varepsilon})$ be the category of Hodge-Tate structures over $\overline{\mathbb{Q}}_{\varepsilon}$, and let…

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

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 field, let $F_{\varepsilon}=F[\varepsilon]/(\varepsilon^2)$, and let ${\mathbb Q}_{\varepsilon}$-mod be the tensor category whose objects are…

Let $DMT(\mathbb Z)$ be the category of mixed Tate motives over $\mathbb Z$, let ${\rm Gal}_{\rm mot}(\overline{\mathbb Q}/\mathbb Q)$ be its motivic Galois groupscheme, and let…

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 $\overline{\mathcal M}_{0,n+3}$ be the compactified moduli space, and let $A$ and $B$ be disjoint unions of boundary divisors. Let $[\Delta'_B]$ be a relative homology class li…

Let $F$ be a number field, let $S$ be a finite set of prime ideals in its ring of integers, and write ${\cal O}_{F,S}$ for the localization of the ring of integers at $S$. Let…

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…

Deligne's conjecture. The weight graded Lie algebra of $U$ is freely generated by

Let $A$ be the cocycle algebra $N$ associated with the Bloch–Kriz construction, let $A\{1\}$ be its 1-minimal model, and let $\operatorname{Conn}^{0f}_{N\{1\}}$ be the category of…