26 problems

Kapaev's asymptotic conjecture. As $x\to\pm\infty$, this solution has asymptotics agreeing with the asymptotics specified in the source's main theorem.

The construction above defines a sequence of functions from radially ordered correlators of Virasoro vertex operators, with parameter $k$ satisfying $\mathsf{b}^2=k$. The resulting…

Fix $u\in B'$ and let $\mathcal{X}:\mathcal{M}'_u\times\mathbb{C}^{\times}\to\mathcal{T}'_u$ be the map satisfying the prescribed wall-crossing jumps, reality condition, and semifl…

Let $J$ be a jump matrix with factorization $J=\Phi_-^{-1}\Phi_+$, let $\tau_W[J]$ be the Widom determinant, let $\tau_{AB}[J]$ be the corresponding Arinkin–Borodin tau function, a…

Matrix Riemann–Hilbert factorization conjecture. The solution is

Uniqueness conjecture. The same growth conditions should guarantee uniqueness of solutions for the RHP for the MBE in this uniformization-variable formulation.

Consider the numerical approximation $u_N$ to the solution $u$ of the Riemann–Hilbert problem on the unit circle, with error estimate … currently established for any $t>2s$. Sobole…

Let $t\in\mathbb{C}^{\times}$ satisfy $\operatorname{Im}(t)>0$. Let $\mathcal{X}_{\beta^{\vee}}(t,\lambda)$ be the specified convergent expression for the conformal limit of the tw…

Solvability and tau-zero conjecture. The Riemann–Hilbert problem for generic $J(z)$ is solvable, and the complex moduli $Q_i[J]$ of the vector bundle are given by the zeros in…

For each topological type $w$ of spectral network, let ${\mathcal V}_w$ be the region where the WKB network has type $w$, let ${\mathcal X}_w$ be the associated coordinates, and le…

Let $\sigma\in\Sigma(Q,W)$ be a generic stability condition, where $(Q,W)$ is the quiver with potential associated to an ideal triangulation of an amenable marked bordered surface…

Let $(\mathbb{S},\mathbb{M})$ be a marked bordered surface, and let … be the mapping class group equivariant continuous monodromy map from the preimage of the dense open set of pro…

Let $Upsilon$ be the Airy Riemann–Hilbert problem solution, let $Omega_1$ denote its first sector, and set $xi:=\mathrm{e}^{2pimathrm{i}/3}$. Let $\operatorname{Ai}$ denote the Air…

Let $\beta>0$, fix $\chi>\frac{2}{\beta}$, and consider the zero-level-line configuration of $\Re(\varphi(\lambda;\chi,\tau))$ in which $\lambda_+(\chi,\tau)$ and…

Let $x$ be a parameter in the elliptic region, and let $\Sigma$ denote the band contour appearing in the associated Riemann–Hilbert problem. As $x$ enters the elliptic region from…

The Novikov equation is formulated through a Riemann–Hilbert problem whose jump matrix has the same form and $k$-dependence as the jump matrix for the Degasperis–Procesi equation.…

Genus-two persistence conjecture. For $|x|>\ln 2$, the solution maintains genus 2 asymptotics for all $t>t_0(x)$ beyond the first break.

Constant-cost computation conjecture. The time required to compute $d_N$ is bounded, the number of Chebyshev coefficients needed to compute the equilibrium measure is bounded, and…

Let $u_0(x)$ be initial data for the Ostrovsky–Vakhnenko equation, and suppose that … for all $x$. A piecewise analytic Riemann–Hilbert problem is expected to arise, with no residu…

Let $G$ and $\hat G$ be the matrix functions in the two vector Riemann–Hilbert problems, and define their indices by … The compatibility conditions relating the solutions of the tw…

Consider the first Painlevé equation $P_I$ and its tritronquée solutions, with $O$ denoting the origin in the complex plane. Dubrovin's conjecture. The tritronquée solutions are an…

Let $\mathcal{Y}^{RH}_{\gamma}(\xi)$ be the piecewise holomorphic function on the $\xi$-plane obtained from the Fock–Goncharov coordinate data, with discontinuities on BPS…

Condition-number estimate conjecture. There exist a Sobolev $W^{s,p}$-norm and a monotone function $\phi$, independent of $(\Gamma,G)$, such that

Pole-location conjecture. The actual positions of the poles are those for which the associated value of $g$ is real and

Let $q(x)$ be a solution of the Painlevé II hierarchy, with Stokes multipliers $s_1,s_2,ldots,s_{2n+1}$ in the associated Riemann–Hilbert problem. Stokes-multiplier restriction. Th…