16 problems

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 $q$ 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…

Kernel inclusion conjecture. There is an inclusion of kernels

Ismail's structure conjecture. Then $(P_n)_{n\geq 0}$ are continuous $q$-Jacobi polynomials, Al-Salam–Chihara polynomials, or special or limiting cases of these families. The same…

Continuous-limit correspondence. If, as $q\to1$, the zeros $\alpha^k_j$ of $\det A(x)$ corresponding to $m^k_j$ satisfy

Quantum A-polynomial conjecture. For any knot $K\subset S^3$,

Let $E$ be an elliptic curve, and let $\operatorname{Diff}_q(X)$ denote the category of $q$-difference modules on a variety or stack $X$ with automorphism $q$. The abelian multipli…

Let $\tau_i$ $(i=1,\dots,6)$ be the tau functions associated with the $q$-Painlevé V equation, with parameters $\theta_*$, $\theta_t$, and $\theta_0$. Write $\overline{f}(t)=f(qt)$…

Noncommutative formal Galois group conjecture. If $q$ is not a root of unity, the injection in Proposition 6.8o is a bijection. Consequently,

Let $A$ be a commutative $L^{\natural}$-algebra, and let the injection in Proposition 6.8m be the functorial map from $\mathcal{CF}_{L/k}(A)$ to the formal group $\hat{G}_3(A)$. Bi…

Let $q$ be a parameter and let the inclusion in Proposition 5.24i be the inclusion of the corresponding formal Galois functors. Equality conjecture. If $q$ is not a root of unity,…

Quantum Galois group conjecture. If $q$ is not a root of unity, the injection in Proposition 6.8o is an equality, so

Equality conjecture. If $q$ is not a root of unity, the injection $\mathcal{F}_{L/k}(A)\rightarrow\widehat{G}_{III}$ is an equality.

Equality conjecture. If $q$ is not a root of unity, the inclusion $\mathcal{NCF}_{L/k}(A)\hookrightarrow\widehat{G}_{II}(A)$ is an equality.

Grothendieck–Katz conjecture for iterative q-difference modules. The iterative $q$-difference module $\mathcal{M}$ is isotrivial, meaning that it becomes trivial after a finite bas…

Let $K$ be the coefficient field, let ${\mathcal P}_f$ be its finite places, and let ${\mathcal C}$ be the cyclotomic places. Suppose that … is a solution of a $q$-difference equat…