39 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…

Let $n$ be an integer with $n\geq 0$, and write $\zeta(\{a\}^n)$ for the multiple zeta value with $n$ successive arguments equal to $a$. A multiple zeta value is called self-dual w…

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 $\mathsf{ConjInv}$ denote the conjugation invariants in dimension $d=3$, and let a shuffle with letters mean a shuffle product involving letters. The signed volume is the degre…

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 $T(\mathbb{R}^d)$ be the tensor algebra, let $V$ be the space of loop invariants, and let $[V,\mathbb{R}^d]$ and $[T(\mathbb{R}^d),\mathbb{R}^d]$ denote the corresponding commu…

Let $T(\mathbb{R}^d)$ be the tensor algebra, let $S$ denote the shuffle algebra of shuffle products of letters, and let $\mathsf{ConjInv}$ denote the conjugation invariants. Define…

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…

For positive integers $k_1,\dots,k_d$, let $\widetilde{T}(k_1,\dots,k_d)$ denote a multiple $\widetilde{T}$-value, and let $V$ be the $\bQ$-linear space spanned by all such values.…

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 $J^{\rm eqv}$ be the generating series constructed in two ways by the constructions referred to in the source as (MGFtoBR63) and (select.1), with expansion parameters…

Equivariant iterated Eisenstein integrals are a construction of non-holomorphic modular forms, and modular graph forms arise in the low-energy expansion of closed-string scattering…

Explicit-description conjecture. These integrals are explicitly described by multiple zeta values and elliptic multiple zeta values.

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 $F\in\mathbb{C}[x,y]$ be a polynomial, let $\gamma\in\pi_1(\Gamma_t,p_0)$ be a loop, and let $k=k(F,\gamma)$ be its orbit depth. For a deformation of $dF=0$, denote by…

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…