46 problems

Let $q$-conformal blocks be the blocks associated with the $q$-difference isomonodromic deformation problem for $q$-Painlevé VI, and let their Fourier transforms define tau functio…

Let a linear differential or $q$-difference system be given, with its Malgrange groupoid acting on the fibers of the associated system. Casale–Roques conjecture. For linear $(q)$-d…

Uniqueness criterion for polynomial solutions. The equation has a polynomial solution only when the parameters $h_j$ are the values specified by $$ and $lambda$ has the displayed i…

Abelianization conjecture. We conjecture that

Conjectural description. The difference Galois group is the smallest algebraic subgroup $G$ of ${\rm GL}(V)$ such that $G$ contains all $G_j$ and the Lie algebra of $G$ contains al…

Let $K_0=k4_3$ be the simplest hyperbolic non-2-bridge knot, and let $J_{K_0}(n)$ denote its colored Jones function. Let $a_j(u,v)\in\mathbb Q(u,v)$ be rational functions with rati…

Birational affine Weyl group action conjecture. The quotient set of such matrices admits a birational action of the affine Weyl group $W(E_8^{(1)})$, with

Let $K\subset S^3$ be a knot, let $F_K(x,q)$ be the two-variable series associated to the sequence of reduced coloured Jones polynomials, and define … Let $\widehat{x}$ and…

Let $q$ be the deformation parameter, $t$ the time parameter, and let $\mathcal{D}$ denote the discrete $q$-lattice … Let $\mathcal{D}$ be the matrix Fredholm determinant defined e…

Let $\tau(t)$ be one of the paper's $q$-isomonodromic tau functions, let $\hat{\tau}_{AB}$ denote a $q$-generalization of an Arinkin–Borodin tau function, and let…

Kernel inclusion conjecture. There is an inclusion of kernels

Ring-morphism conjecture. The map $f\mapsto \mathcal{S}_{q}^{d}(f)$ defines a morphism of rings.

Ring-morphism conjecture. The map $f\mapsto \mathcal{S}_{q}^{d}(f)$ defines a morphism of rings; in particular, it preserves products whenever both sides are defined:

Relative AJ conjecture. For every $\mathcal O\in\mathcal M(\mathcal T)$,

Operator-insertion conjecture. For every $\mathcal O\in\mathcal M(\mathcal T)$: (a) there exists a linear $q$-difference operator $\widehat{A}_{\mathcal T,\mathcal O}$ with a funda…

Holomorphic-block factorization conjecture. For every 1-efficient triangulation $\mathcal T$ there exists a palindromic linear $q$-difference operator $\widehat{A}_{\mathcal T}$ of…

Holonomic-equation conjecture. The state-integral and the rotated 3D-index satisfy identical pairs of linear $q$-difference equations, equal to the homogeneous part of the linear…

Let $I^{\mathrm{rot}}(n,n')(q)$ be the rotated 3D-index, and suppose it satisfies a linear $q$-difference equation in the two variables $n$ and $n'$. Let $A$ denote the $A$-polynom…

Let $\widehat{A}$ be the noncommutative $q$-difference operator associated with a knot, and let $A$ be its classical $A$-polynomial. AJ-Conjecture. Specializing $q=1$ in the…

Colored Jones–state-integral conjecture. The linear $q$-difference equation satisfied by the colored Jones polynomial of $K$ coincides with the linear $q$-difference equations of…

Resurgence and Stokes factorization conjecture. These perturbative series are resurgent, their trans-series involve two non-perturbative variables, and their Stokes automorphism sa…

Let $\kappa\geq 0$, $0<q<1$, and let $a_k\in\mathbb{R}$ satisfy the initial value problem … Here a positive solution means a sequence $\{a_n\}_{n=1}^{\infty}$ with positive coeffic…

Let $K$ be a knot in $S^3$, and let its colored Jones polynomial be regarded as a sequence in the color variable. A $q$-difference equation is an equation involving operators that…

Let $\mathsf S^+(q)$ and $\mathsf S^-(q^{-1})$ be the Stokes automorphism matrices for the transseries of a state integral, represented by $q$- and $q^{-1}$-series, respectively. L…

The perturbative expansions of quantum knot invariants in complex Chern–Simons theory are factorially divergent formal power series. A resurgent function is a formal series whose B…