15 problems

Let $X$ be a simply connected, finite CW complex. It is rationally hyperbolic if the sum of the loop-space Betti numbers … grows at least exponentially in $n$, and let…

Deligne–Drinfeld conjecture. $\mathfrak{grt}_1$ is the completed free Lie algebra with generators $\sigma_3,\sigma_5,\sigma_7,\dots$ of homogeneous weights $3,5,7,\dots$.

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 $H$ be the free abelian group of rank $n$, let $\mathcal{L}_n$ be the free Lie algebra generated by $H$, and let $\operatorname{Der}^+(\mathcal{L}_n)$ be its positively graded…

Constants-equality conjecture. The algebra of constants is exactly $U^{(1)}$:

Hall-monomial constants conjecture. If $M$ is a Hall monomial that is a constant under the derivation $\delta$, then

Term-count conjecture. For $d\geq L$, the maximal possible number of terms satisfies $Q_{L,d}=Q_{L,L}$ and coincides with the number of terms obtained by expanding $R_L$ in the ave…

Optimal recursive relation conjecture. There is no $(2-\epsilon)$-copy recursive formula.

The Grothendieck–Teichmüller Lie algebra is the Lie algebra $D524\mathfrak{grt}_1$; for each $j\geq 1$, let $D524\sigma_{2j+1}$ denote the specified non-trivial odd elements. Let…

Let $F_n$ be the free group of rank $n$, let $\mathcal{L}_n$ be the free Lie algebra associated with the lower central series of $F_n$, and let $\mathrm{Der}^+(\mathcal{L}_n)$ be t…

Pascal descent polynomial conjecture. One has

Odd tabloid solutions conjecture. These two tabloids are the only solutions of $$ with $r+1$ parts when $n=2r+1$.

Tabloid twin and anti-twin conjecture. One has

Twin and anti-twin conjecture outside the support. The words $u$ and $v$ are twin if either $u=v$, or $n$ is odd and $u=\tilde v$; they are anti-twin if $n$ is even and…

Schützenberger's twin and anti-twin conjecture. The words $u$ and $v$ are twin if and only if either $u=v$, or $n$ is odd and $u=\tilde v$; they are anti-twin if and only if $n$ is…