30 problems

The open intersection theory studies intersection numbers on moduli spaces of Riemann surfaces with boundary. Let the open intersection-number generating function be the generating…

Let $Z_N$ denote the Kontsevich–Penner matrix model for $N\geq 1$, and let the refinement be obtained by replacing the complex integral in the refined open partition-function model…

Kontsevich–Penner conjecture. The function $\tau^{o,\mathrm{ext}}_N$ coincides with the tau-function $\tau_N$ given by the Kontsevich–Penner matrix integral.

Pandharipande–Solomon–Tessler's open Virasoro constraint conjecture. The partition function satisfies

Let ${\overline{\mathcal{M}}}_{g,k,l}$ decompose as … where ${\overline{\mathcal{M}}}_{g,k,l,b}$ parameterizes surfaces with boundary having $b$ boundary components. Let $F^{o,N}$…

Fix $r\geq 2$, let $\overline g$ denote the doubled genus, and suppose $b,m>0$. Let the superscript $r\mathrm{BY}$ denote the Bertola–Yang intersection numbers and $r\star$ the num…

Fix $r\geq 2$, and let $\langle\cdots\rangle^{r\mathrm{spin}}_{g,n;b,m}$ denote geometric open $r$-spin intersection numbers, while $\langle\cdots\rangle^{r\star}_{g,n;b,m}$ are th…

Fix $r\geq 2$. Let $\overline g=2g+b-1$, and denote by…

Let $Z_{\rm KP}$ and $Z_{\rm ABT}$ denote the partition functions associated with the Kontsevich–Penner and Alexandrov–Buryak–Tessler constructions. KP–ABT equality conjecture. One…

Let $\langle\tau_{d_1}^{\circ}\cdots\tau_{d_n}^{\circ}\tau_{k_1}^{\partial}\cdots\tau_{k_m}^{\partial}\rangle_{g,n;b,m}$ denote the open intersection numbers with closed and bounda…

Let $Z_{{\rm open}}$ be the partition function of open intersection numbers for bordered Riemann surfaces, defined by … Here $Z_{{\rm closed}}$ is the closed 2-spin partition funct…

Let $g,k,l$ be nonnegative integers, let $d_1,dots,d_l$ be nonnegative integers, and let $\overline{\mathcal{M}}_{2g,k,l,1}$ denote the moduli space of Riemann surfaces with bounda…

Let $\mathcal{Z}({\mathbf t};N)$ be the formal power series obtained from the asymptotic expansion of the Kontsevich–Penner matrix integral, with rescaled times…

Extended refined Kontsevich–Penner conjecture. For any $N$ we have

Geometric boundary-descendent conjecture. There exists a geometric construction of boundary descendents in the refined open intersection theory that gives the extended refined open…

Safronov's refinement conjecture. There exists a refinement of the extended open partition function $\tau^{o,\mathrm{ext}}$ that distinguishes contributions from Riemann surfaces w…

Let $\mathcal{M}_{(g,b),k,l}$ be the moduli space of Riemann surfaces with boundary of genus $g$, with $b$ boundary components, $k$ boundary marked points, and $l$ interior marked…

Let the full generating function range over descendant integrals on moduli spaces of open Riemann surfaces, including all genera and numbers of boundary components. Open KdV hierar…

Q-graded correlator conjecture. These correlators are the $Q$-graded correlators for open intersection numbers. The proposed grading is intended to separate contributions by bounda…

Let $\tau_Q$ be the Kontsevich–Penner matrix-model partition function with integer parameter $Q$, and let $\tau_{\mathfrak{o}}$ be the open intersection-number partition function.…

Pandharipande–Solomon–Tessler's Virasoro conjecture. The partition function satisfies

Let $Q$ be a grading parameter intended to distinguish contributions according to the number of boundary components, and let the resulting generating function refer to the $Q$-refi…

Let the open intersection-number generating function be extended to depend on parameters $S_1,S_2,\ldots$, in addition to the variables for interior marked-point descendants. These…

Let the function introduced in the cited work be the open generating function associated with the Burgers--KdV system. Boundary-descendent conjecture. This function should correspo…

Let the descendent integrals on the moduli spaces of open Riemann surfaces be assembled into their generating series. Open KdV conjecture. In all genera, the generating series of t…