22 problems

Let $N\geq 0$, let $K$ be the field of fractions of $\mathbb{Q}[A,B,C,D,q]/(A-q^{N}D)$, and let $W^{0}$ be the space containing words $(u_{1}v_{1})\cdots(u_{k}v_{k})$ with letters…

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 $X$ be a complex algebraic manifold and let $d$ be a positive integer. Write $QH^k(X)$ for the quotient cohomology groups appearing in the exact sequence associated with iterat…

Shuffle-of-areas closure conjecture. If all $i_j$ are distinct, then

Let $\operatorname{rcl}$ denote the right-closure operator, and let $\operatorname{im}\operatorname{rcl}$ be its image, the space of right-closure invariants. Let…

Let $\mathsf{ConjInv}$ be the algebra of conjugation invariants, and call a homogeneous element a generator at level $n$ if it is one of the homogeneous generators used for this sh…

Let $d$ and $n$ be positive integers, let $C^d_n$ be the cyclic $d$-polytope with $n$ vertices, and let $\mathsf{Inv}^d_{d+2}$ denote the algebra of degree-$d$ volume invariants fo…

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…

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…

Au's conjecture. For all $U(x)\in\mathsf{U}_N$, the renormalized value

Let $G$ be a subgroup of $ame{SL}d(\mathbb{R})$, and let $Z,Z':[0,T]tomathbb{R}^d$ be curves. Suppose that … for every $\varphi\in T(\mathbb{R}^d)$ fixed by $\tilde{\phi}_{A^\top}$…

Let $V_{2n}^{\operatorname{dR}}$ be the $SL_2$-representation appearing above, let $\mathcal{O}(\mathcal{U}_E^{\operatorname{dR}})$ be the Hopf subalgebra generated by Eisenstein w…

Let $\mathcal{A}^{0}$ be the algebra of admissible words, let $L:\mathcal{A}^{0}\to\mathbb{R}$ be the map whose values are multiple zeta values, and let $\mathcal{I}_{{\rm CF}}$ de…

Finiteness conjecture. For any $\gamma$, the torsion-free orbit depth $k$ and the orbit depth $\kappa$ are finite.

Odd-weight alternate odd-position conjecture. Subject to further restrictions on the block lengths and $x$,

Let $B=(0;\ell_1,\ldots,\ell_n)$ be any non-trivial block decomposition of even weight $N$. Alternate odd positions conjecture. … The conjecture extends the analogous alternating i…

Let $X$ be a smooth manifold, let $x_0\in X$, and let $B_s^n(X)_{x_0}^{hom}$ denote the space of homotopy-invariant higher-dimensional iterated integrals of length at most $s$ on b…

Let $G_n(x)$ be the auxiliary polynomial introduced by expanding the expression for $A_{n,2}(x)$, and let $\gamma_n$ be its denominator. Auxiliary-denominator conjecture. For…

Let $A_{n,2}(x)$ be the rational polynomial part of the $n$th iterated integral in the $\ln(1+x^2)$ case, and let $\alpha_{n,2}$ be its denominator. Let $m\ge1$ be an integer. Deno…

Let $A_{n,2}(x)$ be the rational polynomial part of the $n$th iterated integral in the $\ln(1+x^2)$ case, let $\alpha_{n,2}$ be its denominator, and define … Here $p$ denotes a pri…

Deitmar–Horozov conjecture. The restriction map