23 problems

Let $M=[u:d4da0 \rightarrow G^n]$, with $u(1)=(R_1,\dots,R_n)$, be the 1-motive defined by the complex numbers $p_i$, $q_j$, and $t_{ij}$. Choose $p'_1,\dots,p'_{n'}$ and…

Let $X$ be a smooth projective irreducible variety over $\overline{\mathbb{Q}}$, let $\beta$ be the de Rham--Betti comparison matrix in $\mathrm{GL}_N(\mathbb{C})$ under suitable b…

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 $\mathfrak{g}^*$ be the homotopy Lie algebra associated with the free massless theory in dimension $d$, and let $G_{M,\mathrm{Betti}}$ be the motivic Galois group in the Betti…

Let $P$ be the algebra of periods and let $I_{\epsilon_1,\dots,\epsilon_n}$ be the integrals appearing in Drinfeld's associator. Consider the subalgebra of $P$ generated by…

Let $S[BU]=\Sigma^\infty(BU_+)$ be the suspension spectrum of the classifying space of the infinite unitary group, and let $\mathrm{QSymm}_*$ be the Hopf algebra of quasisymmetric…

Let $X$ be a variety, let $M\in\mathcal{ML}(X)$ be a motivic local system, and let $x\in X(\mathbb{C})$. Write $\mathcal{G}_{\textup{mot}}(M,x)$ for the motivic Galois group of $M$…

Let $\mathrm{DMR}_0$ be Racinet's affine group scheme of solutions to the extended double shuffle relations, and let $U^{\rm dR}$ denote the prounipotent part of the motivic Galois…

Motivic Galois group–Grothendieck–Teichmüller isomorphism conjecture. The morphism

Classicality conjecture. The motivic Hopf algebra $\mathcal{H}_{\mathrm{mot}}(k,\sigma)$ should be classical, meaning that it is concentrated in degree zero.

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 $M$ be a 1-motive defined over $\overline{\mathbb{Q}}$. Let $\mathrm{T}_{\mathrm{dR}}$ and $\mathrm{T}_{\mathrm{H}}$ denote its de Rham and Hodge realizations, let…

Triviality conjecture for the defect group. The defect group $P(X)$ should be trivial. Equivalently,

Vologodsky's conjecture. The DG category of log motives should be equivalent to the category of ordinary motives; equivalently, the automorphism group of the Betti functor on log s…

Grothendieck's motivic Galois group conjecture. The preceding motivic and Tannakian description should exist for smooth projective varieties.

Motivic Galois quotient conjecture. The motivic Galois group associated with $\mathcal{H}^{N}$ is a quotient of

Let $(X,\tilde X)$ and $(X',\tilde X')$ be pairs as in the paper, with a morphism $(X',\tilde X')\to(X,\tilde X)$, and let…

Let $K_*=(K_n)_n$ be the tower of motivic groups arising from the moduli spaces $\mathcal{M}_{0,n}$, and let $\mathcal{R}_{\mathrm{Betti}}(K_*)$ and…

Let $D\simeq\mathcal{M}_{0,n_1}\times\mathcal{M}_{0,n_2}$ be a codimension-one boundary component, and let $K^{\bullet}_{\mathbb{Z},n_1,n_2}$ denote the geometric derived group ass…

Over $\operatorname{Spec}(\mathbb{Z})$, let $G^{\bullet}_{\mathbb{Z},n}$ be the affine derived group scheme associated with the mixed Tate category of $\mathcal{M}_{0,n}$, and let…

For two number fields $F$ and $E$, let $\mathcal{M}_{F,E}$ be the category of motives for motivated cycles over $F$ with coefficients in $E$, and let $\mathcal{G}_{F,E}$ be its mot…

Let $E$ be a motive over $K$, and let $G_{\mathcal{M}(E)}$ denote its motivic Galois group. Serre's domination by maximal motives conjecture. After replacing $K$ by a finite extens…

Equivalence conjecture. This functor is an equivalence.