35 problems

Let $S$ be a K3 surface, let $beta eq 0$ be a curve class, and let $n{g,beta}(S)$ denote the corresponding BPS invariant. The class $beta$ has square $beta^2$ and a divisibility in…

Let $\mathfrak{X}$ be a toric Calabi–Yau threefold and let $\hat{\mathfrak{X}}$ be its mirror, whose mirror curve $\mathcal{S}_{\hat{\mathfrak{X}}}$ is defined by … Let…

Let $X=Gr(2,7)[1^{7}]$ be the Calabi–Yau manifold obtained from the Grassmannian $Gr(2,7)$, and let $Y$ be the Pfaffian variety in $\mathbb{CP}^{20}\cap\mathbb{CP}^{6}$. Rødland's…

Let ${\mathfrak X}$ be a toric Calabi–Yau 3-fold, with a Lagrangian brane $L$ and framing parameters $\mathbf t$. Let $\mathcal W_{g,n}({\mathfrak X},L,\mathbf t;x_1,\dots,x_n)$ de…

Let $\mu:M\to\mathfrak{g}$ be proper, let $0$ be a regular value of $\mu$, assume that $\mu^{-1}(0)$ is nonempty, and suppose that $\mathrm{G}$ acts freely on $\mu^{-1}(0)$. For…

Let $Y$ be the blow-up of a smooth projective threefold $X$ along a curve $C$ such that $g(C)\geq 1$, or $g(C)=0$ and $-K_X.C\geq 0$. The associativity equations of the quantum pro…

Let $X$ be a smooth projective variety, let $\{\gamma_a\}$ and $\{\gamma^a\}$ be dual bases of $H^*(X,\mathbb Q)$, and let $g$ and $k$ be integers with $k>g$. Two-point vanishing c…

Let $X$ be a smooth projective variety. Given a basis $\{\gamma_a\}$ for $H^*(X,\mathbb Q)$, let $x_i,y_i\in H^*(X)$ and let $k\geq 2g-3+r+s$. Using Gathmann's conventions … and ……

Quantum puzzle conjecture. The Gromov–Witten invariant $\langle\sigma_\lambda,\sigma_\mu,\sigma_\nu\rangle_d$ is equal to the number of puzzles whose north-west, north-east, and so…

Let $X=G(k,n)$ be a Grassmannian, let $r=n-k$, and fix a degree $d\geqslant 0$. For a partition $\rho$, let $J^d(\rho)$ be the associated $012$-string of length $n$ obtained from t…

Invariant quotient conjecture. For special choices of $\alpha$ and $B$, the invariant $\Phi^{C,k}_{X,B}(\alpha)$ should coincide with a Gromov–Witten invariant of $\hat X$.

Let $Fl_n$ be the complete flag variety, let $S_n$ be its Weyl group, and let $C_{u,v,w}\in\mathbb{Z}[q_1,\dots,q_{n-1}]$ be the polynomials defined by the quantum products of Schu…

Let $X$ be either the Hirzebruch surface $\mathbb F_2$ or the surface $B'$ obtained by blowing up six points on a conic, and let $E$ be its $(-2)$-curve. Let $D$ be an effective di…

Let $(V,\omega,J)$ be a compact symplectic manifold of real dimension $2n$, let $A\in H_2(V,\mathbb Z)$, and let $\overline{\mathcal F}_A(V,g,k)$ be the compactified moduli space o…

Let $X$ be a Calabi–Yau 3-fold, and let its BPS invariants and Gromov–Witten invariants be the enumerative invariants associated with $X$. BPS/Gromov–Witten invariants equivalence…

Let $X$ be a projective Calabi–Yau $4$-fold. Write $\operatorname{Hilb}^n(X)$ for the Hilbert scheme of $n$ points on $X$, let $L\to X$ be a line bundle, and distinguish Gromov–Wit…

Let $Y(q)$ be the Yukawa-type coupling associated with the mirror map of a Calabi–Yau threefold, and let its Lambert series be the Lambert-series expansion obtained from $Y(q)$. Th…

Let $X$ and $Y$ be smooth varieties related by a sequence of flops. Their quantum potentials are functions defined from the corresponding Gromov–Witten invariants, and the Novikov…

Local Fano threefold genus-zero correspondence. One has

Pandharipande–Pixton conjecture. The generating series $\operatorname{PT}(X)$ is conjectured to equal the generating series of Gromov–Witten invariants after a suitable change of v…

Let $X$ be a sextic $4$-fold, let $\beta$ be a curve class, and let $\rho\in H^{4}(X,\mathbb{Z})$. Denote by $\mathrm{GW}_{0,\beta}(\gamma)$ the genus-zero Gromov–Witten invariant…

Let $Y=\operatorname{Tot}_{\mathbb P^1}(\mathcal O_{\mathbb P^1}(l_1)\oplus\mathcal O_{\mathbb P^1}(l_2))$ with $l_1+l_2\geq -1$, and let $d$ be a positive degree. Denote by…

Let $f\in\mathbb Q(d_1,d_2)$ be a rational function defined on $\mathbb Z^+\times\mathbb Z^+$ and suppose … and … for some $k\in\mathbb R^+$. If $n_d$ is a sequence recursively def…

Let $(X,\omega)$ be a compact monotone symplectic manifold with minimal Chern number $\nu$. Let $g,N\in\mathbb Z^{\ge0}$ satisfy $2g+N\ge3$, and let $b_s,c_s\in\mathbb Z^{\ge0}$ an…

Let $(X,\omega)$ be a compact symplectic manifold and let $g\in\mathbb Z$. For fixed classes $H_1,\ldots,H_k\in H^*(X)$, let…