54 problems

Gamayun–Iorgov–Lisovyy conjecture. The tau functions should admit explicit formal series solutions of the form

Let $\tau_{\mathrm{I}}$ be the tau function of the first Painlevé equation, and let an irregular conformal block be a Virasoro conformal block constructed from an irregular vector.…

Higher BGW topological-recursion conjecture. The generating functions of the higher BGW models $\tau^{(m)}$ are given by a version of the Chekhov--Eynard--Orantin topological recur…

The Heisenberg operators $calpha_n$ and Virasoro operators $L_m$ act on tau-functions in the standard $\mathcal{\widehat{\mathfrak{gl}}(\infty)}$ framework. Let…

For each generalized Cartan matrix of affine type $A=(a_{ij})_{i,j\in I}$, let $\tau_i$ be the tau-functions in the Weyl-group representation described above. Strong regularity con…

Scalar-product conjecture. The scalar product of Toda-lattice tau functions is again a Toda-lattice tau function.

Let an $N\times N$ matrix Riemann–Hilbert problem have quasi-permutation monodromies, and let its corresponding tau-function and the appropriately defined determinant of a Cauchy–R…

Let $C({\cal P}_m,{\cal P}_n)$ denote the contact terms associated with the observables ${\cal P}_m$ and ${\cal P}_n$, let $t_n$ be the corresponding time variables, let $V_n$ be t…

Let $\tau_{\mathrm{VI}}^{(0)}(t)$ be the tau function of the sixth Painlevé equation near $t=0$. Let $\theta_0,\theta_1,\theta_t,\theta_{\infty}$ be its parameters, and let…

The paper considers multi-linear forms arising from the relevant blowup relations and the associated Painlevé tau functions. The proposed correspondence identifies the volumes of s…

Painlevé I limit conjecture. There exists a choice of constant $C = C(P_0,{\bf u})>0$ and a polynomial $Q(t_1,t_2,t_5)$ such that, defining $\hat{\tau}_0:= e^{Q}$, considered as a…

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…

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…

Let $\tau(n_1,\ldots,n_s)=\langle g\mid n_1,\ldots,n_s\rangle$ be the discrete function defined for integer tuples with $\operatorname{deg}(\mathbf n)=\sum_i n_i=0$. For…

Let $(\sigma,\eta)$ be the pair of Darboux coordinates characterizing the expansion of the Painlevé $\mathrm{III}_3$ tau function at short distance, and let $(\nu,\rho)$ be the par…

Let ${\cal S}$ be a spectral curve with topological-recursion invariants $F_g({\cal S})$ and $\omega_{g,n}({\cal S})$, and let $\hbar$ be a small parameter. Define the formal tau f…

B-Toda tau-function constraint conjecture. The more general condition

Tau-function Ablowitz–Ladik conjecture. The functions $U$ and $W$ satisfy the positive flows of the Ablowitz–Ladik hierarchy when $t^{2,k}$ is identified with $t_k$ for $k\geq 0$.…

Let the dimer model be coupled to a background connection, and regulate the divergent integrals arising from non-separated zippers as in the referenced section. Let the Jimbo–Miwa–…

Let $B_j$ denote the matrices associated with the punctures, let $epsilon>0$ be the smearing width, and consider the series expansion arising from the regularized determinant of th…

Consider a smeared-out determinant for a singular Fuschian connection, obtained by smoothing the connection with a kernel of width $epsilon$, and expand it as $epsilon\to0$. The ex…

Let $S=(\mathbb{C}\mathrm{P}^1,B_C,\frac{1}{2}z^2,\frac{1}{z})$ be the topological recursion data on the Bessel curve $xy^2=\frac{1}{2}$, and let $\tau_0({\bf t})$ denote the BGW t…

Let $\tau(x,{\bf t},\bar{\bf t})$ be a tau-function of the 2D Toda lattice satisfying the necessary condition … at $t_k+\bar t_k=0$. Higher-flow constrained Toda conjecture. This c…

Kontsevich–Witten hypergeometricity conjecture. The Kontsevich–Witten tau-function is also a hypergeometric solution of the BKP hierarchy.

Generalized BGW hypergeometricity conjecture. The generalized BGW tau-function, after a simple rescaling of times, is a hypergeometric solution of the BKP hierarchy.