18 problems

The Kontsevich matrix model is a matrix-model generating function, and a BKP hypergeometric tau function is a tau function of the BKP hierarchy with hypergeometric coefficients. Al…

Let $\alpha$ be a partition, let $n=l(\alpha)$ be its length, and let $H^g_{\alpha}$ denote the corresponding genus-$g$ Hurwitz number. Let $r$ be the associated number of transpos…

Finite-area equivalence conjecture. The perturbation is equivalent to two-dimensional Yang–Mills theory at finite area, as seen from the contribution of simple Hurwitz space.

Let $G(t,-k)$ denote the generating function considered in the paper, with $t$ a formal variable and $k$ an integer. Generating-function conjecture. For every integer $k\geq 1$, ……

Let $k$ range over positive odd integers and let $\mu$ range over odd partitions. For a spin curve $(B,\vartheta)$, let $H_d^\bullet(B,\vartheta;\mu^1,\ldots,\mu^n)$ denote the dis…

Let $(B,\vartheta)$ be a spin curve, let $g,d$ be integers, and let $k_1,\ldots,k_n$ satisfy … Write $\overline{c}_k=\sum_{\mu\ \textup{odd}}\kappa_{k,\mu}\cdot\mu$ for the spin co…

Let $g>0$ and let $\vec{\ell}\in\mathbb{Z}_{\geq0}^n$ satisfy $|\vec{\ell}|\leq g-1$, where $|\vec{\ell}|$ is the sum of the entries of $\vec{\ell}$. Let $D_{(\vec{\ell},i),2g+2}$…

Nonnegativity and log-concavity conjecture. The integers $c_{\vec{i},k}$ and $\widetilde{c}_{\vec{i},k}$ are nonnegative, and, for fixed $\vec{i}$, the sequences…

Let $\vec{i}$ be a tuple of nonnegative integers, with $|\vec{i}|$ denoting the sum of its entries. Let $D_{(\vec{i}),2g+2}$ and $d_{(\vec{i}),2g+2}$ denote the corresponding…

Let $H_{g,n}(x_1,\dots,x_n)$ be the $n$-point generating function for connected $q$-orbifold monotone Hurwitz numbers, and consider the spectral curve … Let…

Let $g\geq 0$ and $n\geq 1$. For positive integers $\mu_1,\ldots,\mu_n$, let $h_{g;\mu}^{\circ,q,r}$ be the connected $q$-orbifold $r$-spin Hurwitz number, and define the formal sy…

Radius-of-convergence conjecture. The radius of convergence of $\vec{\mathbf{F}}_g(z)$ is exactly

Let $\overline{P}(\mathbf{x})$ be the moduli space appearing in the proposed ELSV formula for double Hurwitz numbers, and let $\Lambda_{2g}$ be its tautological Chow class of degre…

Let $\mathbf{x}=(x_1,\ldots,x_n)\in\mathbb Z^n$ satisfy $\sum_i x_i=0$, and let $H_g(\mathbf{x})$ denote the corresponding double Hurwitz number. Let $\overline{P}(\mathbf{x})$ be…

Let $\mathbf{x}=(x_1,\ldots,x_n)\in\mathbb Z^n$ satisfy $\sum_i x_i=0$, and let $H_g^r(\mathbf{x})$ be the double Hurwitz number, viewed piecewise polynomially on the chambers of t…

Let $g$ be fixed and let $\eta=(\eta_1,\ldots,\eta_n)$ be a ramification profile with $n=\ell(\eta)$. Set $r=2g-2+d+n$, where $d=\sum_i\eta_i$, and let $H_g(\eta)$ denote the corre…

Hurwitz theory is defined using the partition function … where $Q={\rm e}^{-t_H}$, $g_H$ is the string coupling constant, $t_H$ measures the size of the target ${\mathbb P}^1$, and…

Let $Z(t_H,g_H)$ be the Hurwitz partition function, with free energy … where $Q={\rm e}^{-t_H}$ and the coefficients encode simple Hurwitz numbers. The matrix-integral conjecture f…