29 problems

Consider a classical confluenting family of hypergeometric equations, with a nonperturbed equation having an irregular singularity. Let the perturbed equation have appropriately no…

An irregular singularity of a linear differential equation has Stokes operators, and a deformation of the equation has monodromy whose limiting behaviour can be considered as the d…

Fix $u_1,\ldots,u_n\in{\mathds C}$, $\xi\in S^1$ with $\Re((u_i-u_j)/\xi)<0$ for $i<j$, and $T\in M(n\times n,{\mathds R})$ upper triangular with $T_{ii}=1$ and $T+T^{tr}$ positive…

Let $(w,u_1,\ldots,u_n,\xi,T)$ be data with $w\in{\mathds Z}$, $u_i\in{\mathds C}$ pairwise distinct, $\xi\in S^1$ satisfying $\Re((u_i-u_j)/\xi)<0$ for $i<j$, and…

Let $(H,\nabla,H'_{\mathds R},P)$ be a TERP-structure. It requires no ramification when its formal decomposition can be made without ramification; it is a mixed TERP-structure when…

Let $(\nu^\pm,\rho^\pm)$ be two parameter pairs, and let $\mathscr T^\pm(t,\nu^\pm,\rho^\pm,\hbar)$ be analytic $\tau$-functions of $(P_{\rm I})$ defined on domains containing…

The discrete Airy equation has an infinite-parameter transseries whose exponential contributions produce accumulations of Stokes and anti-Stokes curves. Generic accumulation conjec…

The model under consideration has a Stokes constant and generalized DT invariant $n_{0,1}$ associated to the singularity $X^1$; the table gives the value $n_{0,1}=7884$. Stokes-con…

Let $\nabla$ be a WKB-regular $\hbar$-flat connection with strongly GMN spectral curve, let $\Psi$ be a WKB formal solution, and let $\cL\Psi$ be its Borel–Laplace dual. Let…

Let $\widehat{\mathfrak{G}}^0$ be the subgroup generated by the periods and Stokes jumps obtainable from a chosen initial integral $I^{\bbalpha_0}(k)$, and let $\mathfrak{G}_S$ den…

Zeta-content conjecture. In the large-$g$ limit, all transcendental numbers contributing to the Stokes data appear only through the digamma functions, and only through their zeta-f…

Digamma transcendence conjecture. The transcendental part of the Stokes data comes uniquely from the digamma functions.

Let $\alpha$ and $\beta$ be critical points, let $\tau(\alpha)$ and $\tau(\beta)$ denote their corresponding torsions, and let $\mathcal{S}_{\alpha}^{\beta}$ be the Stokes coeffici…

Let $\widehat{\Phi}(h)$ be the matrix of completed formal asymptotic power series associated with boundary-parabolic $\operatorname{PSL}_2(\mathbbm C)$-representations, and let…

Let $N^{(1)}_{1-x}$ be the coefficients in the first diagonal of the Stokes data, and write their asymptotic completion in the form … Here the coefficients $a_i$ are correction ter…

Vanishing conjecture. For every higher diagonal $\ell>1$, all limits $\lim\limits_{g\to+\infty}d^{n}_{\ell}(g)$ vanish for every $n>0$. This is supported by explicit checks for the…

Stokes matrix–rotated 3D-index conjecture. For every hyperbolic knot $K$,

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…

Consider the cubic equation with spectral covering $\Sigma=\{y^3+p_2(z)y+p_3(z)=0\}$, with simple branch points, and let $\psi_i^\vartheta$ be the local WKB solution associated wit…

Fix $(t,\nu)\in D_{t_\ast}\times D_{\nu_\ast}$ so that the Stokes graph has no Stokes segment. Let $C$ be a Stokes curve connecting a branch point $e$ and $\infty$, forming a commo…

Let $M>1/2$, let $k$ be any value, and let $\psi^{(1)}_{[k]}(z,\overline{z};\theta)$ denote the Symanzik-rescaled solution of the differential equations for $\psi^{(1)}$ and…

Let $Q(z,\eta;t)$ be a regular deformation of Schrödinger potentials satisfying the listed polynomiality, analyticity, pole-order, and regularity conditions, and let $W_{\beta}(t)$…

Let ${\bf c}$ be the parameter of $(P_{\rm III'})_{D_{6}}$, and let the walls be the specified loci in parameter space across which the Stokes geometry may degenerate. Consider the…

Let $R_{\rm odd}$ be the odd part of the formal WKB solution, and let $\lambda(t,{\bf c},\eta;\alpha)$ be the corresponding transseries solution of $(P_{\rm III'})_{D_{6}}$, with f…

Fix an arbitrary point $\overline{c_0}\in {\mathbf E}_q\setminus\Sigma_{A_0}$. For a $q$-difference system $A$, let $\Delta_\alpha^{(\delta,\overline{c})}$ denote the corresponding…