32 problems

For the Deligne–Mumford moduli space of stable curves $\overline{{\mathcal M}}_{g,h}$, let $\Lambda_g^\vee(u)$ denote the total Chern polynomial of the Hodge bundle and let…

Two-partition Hodge-integral formula. One has

Let $\overline{\mathcal{M}}_{g,k}$ be the moduli space of stable algebraic curves, let $\mathbb{E}_{g,k}$ be its Hodge bundle, and let $\mathcal{C}_g(z)=\sum_{j=0}^g\lambda_j z^j$…

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

Let $G^{\bullet}(\lambda;x,y,-x-y;p^+,p^-)$ be the exponential generating function built from the two-partition Hodge integrals $G_{\mu^+,\mu^-}$, and let…

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 $\theta$…

The $\lambda_g$-conjecture.

Sine formula conjecture. For all $d\geq 1$,

Let $n$ be a positive integer, let $p$ be the parameter appearing in the generating function $C_{n,1}(\lambda;p)$, and let $\lambda$ be the formal parameter. Zhou's one-leg conject…

Let ${\mathcal P}_+$ denote the set of nonempty partitions, let $\mu^+$, $\mu^-$, $\nu^+$, and $\nu^-$ be partitions, and let $G_{\mu^+,\mu^-}(\tau)$ be the specialization defined…

Let ${\mathcal P}$ be the set of partitions, let $\mu^+$ and $\mu^-$ be partitions, and let $G_{\mu^+,\mu^-}(x,y)$, $p^+_{\mu^+}$, and $p^-_{\mu^-}$ denote the corresponding genera…

The $\lambda_g$ conjecture. The Hodge integral satisfies

Virasoro conjecture for Hodge integrals. For any smooth projective variety,

Let $X$ be a smooth projective variety, and let $\Psi_{g,n}$ denote the universal expression for Hodge integrals with one $\lambda_{g-1}$ or $\lambda_g$ insertion defined in the so…

The $\lambda_g$ formula. One has

Let $X$ be a smooth projective variety. Let $F_g$ be the genus-$g$ generating function, and write $\langle\langle\cdots;\lambda_g\rangle\rangle_g$ for the corresponding Hodge-integ…

Let … is a consequence of the degree-$g$ topological recursion relations proposed in the cited work. This proposed connection relates universal equations for higher-genus Gromov–Wi…

Let $a=(a_1,\ldots,a_n)$ be a vector of non-negative integers, and let $P_a(\alpha,t)$ be the power series defined by the double Hodge integrals in the paper. The notation $|a|$ de…

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 $\tau_{KW}$ denote the Kontsevich–Witten tau-function and $\tau_H$ the Hodge tau-function. Both are KP tau-functions, hence points in the Sato Grassmannian related by an elemen…

Let the Hodge tau-function be the KP tau-function generating linear Hodge integrals, and let the Kontsevich–Witten tau-function be denoted by $\tau_{KW}$. The corresponding point o…

Let $Z_{q,p}({\bf T})$ be the generating function of triple Hodge integrals … with … The parameters satisfy the Calabi–Yau condition, and $\mathbf T$ denotes the Hodge-integral var…

The topological vertex is a combinatorial object used to compute all-genus A-model topological string amplitudes on smooth non-compact toric Calabi–Yau threefolds. Li, Liu, Liu and…