19 problems

Let … where $i$ is an integer, and let $s$ be the unique formal power series in $x$ and $p$ satisfying … For the Hurwitz generating series $H_g(x,p)$, and for a partition $\theta$…

For $m\ge 1$, let $e_i$ denote the $i$-th elementary symmetric function in the variables used to define $f_m^{(1)}$. The closed formula conjecture for genus-one ramification polyno…

For $g=0$ and $m\ge 3$, or $g\ge 1$ and $m\ge 1$, let $f_m^{(g)}(\alpha_1,\ldots,\alpha_m)$ denote the rescaled ramification number associated with coverings of the sphere by a gen…

Let $m\ge 1$, let $\alpha=(\alpha_1,\ldots,\alpha_m)$ be a ramification type, and let $n=\alpha_1+\cdots+\alpha_m$. For $i=1,\ldots,m$, let $e_i$ be the $i$-th elementary symmetric…

Hurwitz-number congruence conjecture. If $d$ is not divisible by $4$, $6$, or $9$, then

Do–He–Robertson's coefficient-gap conjecture. The coefficients satisfy

Let $d\geq10$ and let $\nu$ be a partition of $d$ with $m_1(\nu)\geq2$. Let $z_\nu$ be the standard centralizer factor, and let $b^X_\nu(\mu^{(1)},\dots,\mu^{(s)},m)$ denote the co…

Let $d$ be the degree, let $\boldsymbol{\nu}$ be the ramification data, and let $\operatorname{logDR}_0(d,-\boldsymbol{\nu})$ denote the logarithmic double ramification cycle. Let…

Let $d$ be the degree, let $\boldsymbol{\nu}$ be the ramification data, and let $\operatorname{logDR}_0(d,-\boldsymbol{\nu})$ denote the logarithmic double ramification cycle. Let…

Let $r>2$ and let $Q_1,\ldots,Q_r$ and $\Lambda$ be the parameters of the half Seiberg–Witten curve, with $B_{2g}$ denoting the Bernoulli number. Topological free-energy conjecture…

Fix positive integers $\mu_1,\ldots,\mu_n$ whose sum is $d$. Let the roots of the polynomial $\vec{H}^t_{g,n}(\mu_1,\ldots,\mu_n)$ be ordered from smallest to largest. Genus-asympt…

Let $g\geqslant 0$, $n\geqslant 1$, and let $\mu_1,\ldots,\mu_n\geqslant 1$. The deformed monotone Hurwitz number $\vec{H}^t_{g,n}(\mu_1,\ldots,\mu_n)$ is a polynomial in $t$. Two…

Frobenius-manifold Lyashko–Looijenga degree conjecture. The degree of the map $LL(\mathcal{F}_g)$ equals

Let $W=(NQ)^{-1}$ be the Wishart matrix described above, and define $c_0(\mu_1,\dots,\mu_\ell)$ by … Here $\ell\geq 1$, $(\mu_1,\dots,\mu_\ell)\in\mathbb{N}^{\ell}$, and $C_\ell$ d…

Orbifold Zvonkine formula. The $q$-orbifold $r$-spin Hurwitz numbers satisfy

Let $d$ be a positive integer. For a partition $\alpha$ of $n$, let $h^d(\alpha)$ denote the minimal $d$-Hurwitz number, let $\mu^d(\alpha)$ denote its minimal factorization length…

Let $g\ge 0$ and let $\mu$ be a partition with $|\mu|\ge 3$. Write $h_{g,\mu}$ for the corresponding simple Hurwitz number, $r(g,\mu)$ for its associated ramification number, and…

For each $g,n$, let $H_{g,r}(x_1,\dots,x_n)$ be the generating function formed from the invariants $h_{g,r;k_1,\dots,k_n}$, and let $D$ denote the differential operator … Let…

Let $r$ be a positive integer. For integers $k_1,\dots,k_n$, write … where $p_i$ is the quotient and $r-1-a_i$ is the remainder on division by $r$, and set … Let…