34 problems

Polynomial lower-bound conjecture. For given $n\ge 1$ there exist constants $c(n)>0$ and $\lambda(n)>0$ such that

Let $q=\exp(\pi i/\ell)$ with $\ell\ge3$, and let $\operatorname{Conf}_{\vec\Lambda}^{(\ell)}$ be the space spanned by the lattice restrictions of the Weyl anti-symmetrized hyperge…

Rational-function lower-bound conjecture. If

Let $Z$ be an infinite subset of the set of all roots of unity over $\mathbb{Q}$, and let … be the homomorphism induced by evaluating polynomials at the elements of $Z$. Injectivit…

Let $N>1$, and consider multiple polylogarithms evaluated at $N$-th roots of unity. The double-shuffle and distribution relations are families of relations among these values. Prim…

Safronov's root-of-unity skein conjecture. There is a line bundle $\mathcal{L}_q$ on $\mathrm{Loc}_{G^\vee}(M)$ such that

Let $\omega$ be a primitive $p$-th root of unity with $p\geq 1$, let $K$ be a nice knot, and let $F_K(x,q)$ be its $F_K$ series. Let $P_{K,\omega}^{(j)}(x)$ be the polynomials occu…

Let $K$ be a knot, let $J_K$ be its colored Jones function, let $\omega$ be a primitive $p$-th root of unity with $p\geq 1$, and let $\Lambda$ be the source's ring containing $J_K$…

Let $K$ be a nice knot, let $F_K(x,q)$ be its $F_K$ series, let $N_p(K,x)$ be its ADO polynomial, let $\Delta_K(x)$ be its Alexander polynomial, and let $\zeta_p=e^{2\pi i/p}$ be a…

Let $X$ and $X^!$ be symplectically dual varieties arising from Coulomb-branch and resolved Higgs-branch constructions for the gauge theory $(G,N)$. Let $\zeta$ be the relevant roo…

Let $X=\{x_0=1,x_1,\ldots,x_k\}$ be a set of pairwise distinct roots of unity, and let $B=\{b_0=0,b_1,\ldots,b_m\}$ be a set of coprime integer numbers. Let $M=(x_i^{b_j})$ be dege…

Let $k$ be even, let $\zeta$ be a primitive $2k$-th root of unity, and let $Q_k$ denote the set of quasipolarities. For each permutation $\pi\in Q_k$, consider the product over…

Let $m$ and $n$ be positive integers, let $q$ be a nonzero complex number, and let $Q_{m,q}(n)$ and $Q_m(n,\mathbb{C})$ denote the $q$-deformed and ordinary $m$-quasi-invariant pol…

Pilehrood–Pilehrood–Tauraso's conjecture. For any positive integer $n$ with $n>k$, one has

Let $n>1$ be an integer, let $\zeta$ be a primitive $n$-th root of unity, and define the derangement set … For a permutation $\tau$, let $\operatorname{sign}(\tau)$ denote its sign…

Assume $n=3$, $\gcd(a_1,a_2,a_3)=1$, and let $\mathcal{A}_p=\{a_1\bmod p,a_2\bmod p,a_3\bmod p\}$ for $p=2,3$. Let $\zeta$ be a primitive cube root of unity, and let…

Cyclic-sum conjecture. For every integer $n>r$ and every primitive root of unity $\zeta_n$, the following hold. With $r=\sum_{j=0}^{t}d_j+2t$,

Let $DS_n$ be the set of complex numbers that occur as eigenvalues of $n$-by-$n$ doubly stochastic matrices, and let $PM_n$ be the union of the convex hulls of the $k$-th roots of…

Let $B^6=\{B^6_n,n\geq1\}$ be the Broadhurst set, where the sets $B^6_n$ are formed from iterated integrals indexed by Lyndon words in the letters $0$, $\xi_3^2$, and $\xi_2=-1$, s…

A regular matrix $H\in M_N(\mathbb C)$ is a complex Hadamard matrix whose row scalar products decompose as sums of cycles, and an affine deformation is the deformation notion used…

A partial complex Hadamard matrix $H\in M_{M\times N}(\mathbb T)$ is regular when the scalar products between its rows decompose as sums of cycles. A Butson matrix is a complex Had…

Let $\mu_{2k+1}$ denote the set of $(2k+1)$-th roots of unity. For every prime factor $p$ of $2k+1$ and $\beta_p\in\left[0:\frac{2k+1}{p}-1\right]$, let … be the corresponding $p$-…

Let $n$ be a positive integer, let $f:\mathbb{Z}_n\longrightarrow\mathbb{Z}_n$ be a function, and let $\mu_f^{a,b}$ denote the average associated with $f$ for $a,b\in\mathbb{Z}_n$.…

Let $G$ be a primitive log-divergent graph with loop number $h_G$ and $c_2(G)=-z_3$. $c_2=-z_3$ conjecture. … The three known examples lie in the indicated space, and the source ob…

Let $G$ be a primitive log-divergent graph with loop number $h_G$ and $c_2(G)=-z_2$. $c_2=-z_2$ conjecture. … Here $\mathcal{Z}_2$ is the algebra of multiple polylogarithmic period…