23 problems

Let ${\mathfrak h}^{\mathbb Q}_{g,1}$ be the rational graded Lie algebra of symplectic derivations, let $H_{\mathbb Q}$ be the rational symplectic module, and let…

Let $\mathfrak h_{g,1}^{\mathbb Q}$ be the graded Lie algebra under consideration, and let $\operatorname{Sp}$ act on it through the symplectic action on $H_{\mathbb Q}$. Define…

Let $\mathbb P^1-\{0,1,\infty\}$ be the thrice-punctured projective line over $\mathbb Q$, let $Gal(\overline{\mathbb Q}/\mathbb Q)$ be its absolute Galois group, and let…

Let $T=\mathbb{P}^{1}_{\mathbb{Q}}-\{0,1,\infty\}$, and let $\mathfrak{t}=\bigoplus_{m>0}\mathfrak{t}_m$ be the graded Lie algebra over $\mathbb{Z}_p$ associated with the lower cen…

Let $X$ be the once-punctured elliptic curve under consideration, let $K$ be its complex multiplication field, let $I_K$ be the specified index set, and let $\boldsymbol{m}$ range…

Let $\mathfrak{g}$ be a complex finite-dimensional Lie algebra, and let $(\mathcal{M},\delta)$ be an abelian magma such that $\mathfrak{g}$ is $\mathcal{M}$-magma-graded, with the…

Let $I$ be the index set of multi-indices used to define the characters $\kappa_{\boldsymbol{m}}$, let $g$ be the associated graded Lie algebra, and let $\chi^{\boldsymbol{m}}$ denot…

Let $\xi_B$ and $\xi_D$ be the block-graded and depth-graded functionals associated with the relations in the preceding block-to-depth reduction theorem, and let $\sigma$ be an ele…

Depth-to-even-block isomorphism conjecture. The map ${\mathfrak{s}}{\mathfrak{d}}{\mathfrak{g}}\to{\mathfrak{e}}{\mathfrak{b}}{\mathfrak{g}}$ is an isomorphism. It is stated to hol…

Let ${\mathfrak{d}}{\mathfrak{g}}$ be the depth graded motivic Lie algebra, let ${\mathfrak{s}}{\mathfrak{d}}{\mathfrak{g}}$ be the subalgebra generated by the non-exceptional elem…

Let ${\mathfrak{d}}{\mathfrak{g}}$ be the depth graded motivic Lie algebra, and let $\mathrm{S}$ be the space of cusp forms for $\mathrm{SL}_2(\mathbb{Z})$. Homological Broadhurst–…

Let $L$ be a Lie algebra with a group grading $\Gamma$ that is a coarsening of an abelian grading of $L$. Abelianness conjecture. Then $\Gamma$ is abelian. The paper states that no…

Let $\operatorname{Gr}\operatorname{IA}_n$ be the graded Lie algebra associated to the Andreadakis filtration of $\operatorname{IA}_n$. The stable Lie-algebra comparison conjecture…

Let $G$ be the connected reductive group with involution $\theta$, let $K=G^\theta$, and let $\mathfrak g_1$ be the degree-one component of the associated graded Lie algebra. For a…

Let ${\mathfrak{g}}_1$ be the degree-one component of one of the stably graded Lie algebras considered here, let $K$ act on ${\mathfrak{g}}_1$, and let ${\mathfrak{g}}_1^{rs}$ be i…

Let $g$ be the genus, let $\mathfrak{t}_g$ be the relevant graded Lie algebra associated to the Torelli group, and let $\mathfrak{m}_g$ be the Lie algebra of symplectic derivations…

Let $H(4,(m_1,m_2,m_3,1))$ and $S(3,(m_1,m_2,m_3))$ denote the indicated graded Lie algebras of Cartan type in characteristic $2$. Brough–Eick's exceptional-isomorphism conjecture.…

Extra-special orbit conjecture. The number of $\mathfrak X_{\Delta(1)}$-orbits is $\#\Pi_l$, every orbit has size $h-1$, and, if $h$ is even, every orbit contains a unique Lagrangi…

Orlik's orbit-average conjecture. The following assertions hold: (i) ${\mathsf{ord}}(\mathfrak X_{\Delta(1)})=d_1+1$; (ii) the average size of the antichains in every…

The t=-1 phenomenon conjecture. The value $\EuScript M_{\Delta(1)}(-1)$ equals the number of upper ideals $I$ satisfying $I^*=I$.

Generalized M-polynomial product conjecture. One has

Let $\mathcal{I}_{g,1}$ be the Torelli group of a genus-$g$ surface with one boundary component, and let $\operatorname{gr}(\mathcal{I}_{g,1})$ be the graded rational Lie algebra a…

Let $H$ be a distribution with co-rank $s$ and rank $r$, so that its derived distribution satisfies $[H,H]=T$ and hence $s<r(r-1)/2$. Square-root growth conjecture. The minimal pos…