18 problems

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

Let ${\cal S}$ 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…

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 $k\geq 0$.…

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…

Let $\operatorname{DP}$ be the set of strict partitions, and let a hypergeometric BKP tau-function be expressed as … with $r_\lambda=e^{\sum_{j=1}^{\ell(\lambda)}\xi(\lambda_j)}$.…

Let $\operatorname{DP}$ be the set of strict partitions. A hypergeometric BKP tau-function has the form … where $r_\lambda=\prod_{j=1}^{\ell(\lambda)}\prod_{m=1}^{\lambda_j}r(m)$ f…

Let $\operatorname{DP}$ denote the set of strict partitions, let $Q_\lambda({\bf t})$ be the normalized Schur Q-function, and let $\delta_{k,j}$ denote the specialization with valu…

Let $u=q^{2\sigma}$ and let $\tau$ and $\uptau$ denote the $c=1$ and $c=-2$ Painlevé VI tau functions, respectively. Let $\td{u}=\td{u}(u)$ and $\td{s}=\td{s}(s,\sigma)$ be functio…

Let $\tau(\boldsymbol{\theta};\sigma,s|t)$ be the tau function defined from the five-dimensional Nekrasov partition function, and let…

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 $\overline{f}(t)=f(qt)$…

Let $\mathcal M_{\mathcal W({\mathfrak{sl}_d})}^{\mathbb P^1}$ be the moduli space of marked points, weights, and times, and let $\mathfrak T_{\bold z,\bold \alpha,\bold t}$ be the…

Let $q(t\,|\,\boldsymbol\nu)$ be a Painlevé I function with Stokes parameters $\boldsymbol\nu$, and let $\mathcal R_k$ be a ray on which $t\to\infty$. Define … Let $G(z)$ denote th…

Heisenberg–Virasoro extension. The KW and Hodge tau-functions are related by

Long-distance expansion conjecture. The sine-Gordon/Painlevé $\mathrm{III}_3$ tau function is given by the convergent series

Let $S(\lambda,x)$ be the vessel matrix and let $\tau(x)$ be the associated tau-function. The entries of $S(\lambda,x)$ are understood with respect to some $p$-norm on $\mathrm I$.…

Let $F(v)$ be a solution of the WDVV equations, let $\hat F(\hat v)$ be its inversion, and let $\tilde{\mathcal F}_g$ and $\hat{\mathcal F}_g$ be the corresponding genus-$g$ free e…