40 problems

Let $K\subset S^{3}$ be a knot, let $L_{K}$ be the corresponding Lagrangian in the resolved conifold $X$, and let $A_{K}(x,p,Q)$ be the augmentation polynomial of $K$, where $x$,…

Aganagic–Vafa mirror conjecture. The superpotential agrees with the disk potential $F^{\mathcal{X},(\mathcal{L},f)}$ up to a mirror transform; equivalently, the disk potential agre…

Let $X$ be a semi-Fano toric manifold, meaning that its anti-canonical divisor $-K_X$ is nef. Let $W^{\text{HV}}$ and $W^{\text{LF}}$ denote its Hori–Vafa and Lagrangian Floer supe…

Let $t_1,t_2$ be the flat coordinates defined by the logarithmic solutions of the relative Picard–Fuchs system, and let $W(t_1,t_2)$ be the superpotential expressed in these coordi…

Consider the noncompact relative Picard–Fuchs system with moduli coordinates $z_i$, arising from the relative cohomology of the pair consisting of the hypersurface and its divisor.…

Let $S$ be a moduli specification, let $\vec{a}$ be a vector of nonnegative integers, and let $\vec{\gamma}$ be a vector of cohomology classes in $H^*_T(\mathbb{P}^1)$. The combina…

A Looijenga pair is a rational smooth projective surface $S$ together with an anticanonical singular nodal curve $D_1+\cdots+D_l$. Its logarithmic Gromov--Witten invariants are den…

Let $\Gamma\subset S^2$ be a cubic planar graph, let $L_\Gamma\subset J^1(S^2)$ be the associated Legendrian surface, and let…

Let $K$ be a framed, oriented knot, let $W_R(K)$ denote its gauge-theory invariant in ChernSimons theory, and let … be the resolved conifold with Lagrangian boundary condition…

The q-difference equations obtained from the $U(1)$ skein specialization of the quantum mirror operator are associated with a Legendrian boundary admitting a Lagrangian filling, wh…

Let $(X,D)$ be a nef Looijenga pair admitting a deformation to a pair $(X',D')$ with $X'$ toric, $D'_i$ prime toric divisors for $i<l$, and $D'_l$ nef. Let $(Y,L)$ be the correspon…

Let $(X,D)$ be a nef log Calabi–Yau surface pair with $D=D_1+\cdots+D_l$, and let $(Y,L)$ be the associated toric Calabi–Yau threefold with framed Aganagic–Vafa branes…

Let $Y(D)$ be a log Calabi–Yau surface with maximal boundary, let $d$ be a $D$-convex curve class, and let $Y^{\rm op}(D)=(Y^{(l)}\setminus D^{(l)},L)$ be the associated open geome…

Let $S$ be a moduli specification, and let $\mathrm{OGW}(S,\vec{a},\vec{\epsilon})$ denote the associated open Gromov–Witten invariant. Call $(S,\vec{a},\vec{\epsilon})$ underdeter…

Let the higher-genus open stationary descendents be the proposed open invariants of $\mathbb{CP}^1$ with stationary insertions, defined using a coherent boundary condition. Geometr…

Let $\Gamma$ be a localization graph, let $\mathcal{F}_{\Gamma}$ be the fixed-point component parameterized by $\Gamma$, and let $e_{\Gamma}^{-1}$ be the class defined in the sourc…

Open Gromov–Witten integrality conjecture. The coefficients $n_{0,Q,m}$ are integers and count BPS states in class $(Q,m)$. The expansion is the resummation of multicover contribut…

Open Crepant Transformation Conjecture. Under analytic continuation in the moduli parameters, the disk potentials for the corresponding Aganagic–Vafa branes should be identified.

The Gross–Siebert construction associates wall-crossing functions to a mirror family, and open Gromov–Witten invariants are generating functions counting holomorphic curves with su…

Let $F^o$ be a solution of the WDVV equations satisfying the unit and homogeneity properties, together with its calibration. Under possibly additional assumptions, let…

Let $\mathcal{F}^{\mathrm{pt},o}_g$ be the genus-$g$ open generating series arising from the intersection theory on moduli spaces of stable pointed disks, with…

The remodeling conjecture. The open-closed Gromov–Witten potentials satisfy the displayed identities relating them to the topological-recursion invariants, including the stable cas…

Let $K\subset S^3$ be a knot, let $X$ be the resolved conifold, and let $L_K\subset X$ be the Lagrangian conormal of $K$ shifted off the zero section. HOMFLY–PT/open Gromov–Witten…

Consider the topological A-model on a local toric Calabi–Yau 3-fold $Y_{\bm{l}}$ with a special Lagrangian submanifold $L$. Let $F_n^{(g)}(\mathsf{x}_1,\ldots,\mathsf{x}_n)$ be the…

Let $X=X_\Delta^3$ be a Calabi–Yau 3-fold and let $L=L_{AV}$ be an Aganagic–Vafa A-brane. Let $X^\vee\subset(\mathbb{C}^\times)^2\times\mathbb{C}^2$ be its mirror, and let $W$ be t…