30 problems

Let $X$ be the Calabi–Yau threefold considered above, let $\beta\in H_2(X,\mathbb Z)$, and let … be the surjective morphism from the normalization of the sheaf moduli space to the…

Let $C$ be an integral projective locally planar curve over $\mathbf{k} = \mathbb{C}$, let $g=g(C)$ and $\widetilde{g}=\widetilde{g}(C)$ be its arithmetic and geometric genera, and…

Let $(C,0)$ be a reduced planar curve singularity over $\mathbf{k} = \mathbb{C}$, let $\delta$ be its $\delta$-invariant, and let $n_h(C,0)$, $0\leq h\leq\delta$, be the BPS invari…

Let $A$ be the contraction algebra of a rational flopping curve $C$, namely a Jacobi algebra of the type defined in the cited work. For every $d\in\mathbb{N}$, let…

Let $S$ be a smooth projective surface, let $\beta\in H_2(S,\mathbb{Z})$ satisfy … and let $K_S$ be the total space of the canonical line bundle over $S$, viewed as a non-compact C…

Consider a type II$_{b}$ limit associated with a Tyurin degeneration of the form $V_0=V_1\cup_Z V_2$ and a special Lagrangian 3-cycle $\Gamma_0$ dual to an element of…

Let $\mathcal{L}^{\vec{\tau}}$ be a framed $L$-component link with component framings $\vec{\tau}=(\tau_1,\ldots,\tau_L)$. For nonnegative integers $r_1,\ldots,r_L$, not all zero,…

Let $S$ be a rank $2$ local snc del Pezzo surface, let $D$ be a polarization, and let $N^0_\gamma$ and $n^{0,D}_\gamma$ denote its local genus-zero Gromov–Witten and local genus-ze…

Let $S$ be a Gorenstein surface with ample anticanonical sheaf, let $D$ be a polarization of $S$, and let $n^D_\gamma$ be the local BPS invariant associated with a curve class…

For a symplectic log Calabi–Yau fourfold $(X,D)$, let $A\in H_2(X,\mathbb Z)$ and $g\geq 1$. Define … where $\mu$ is the Möbius function, $q=\operatorname{e}^{\mathrm{i}h/2}$,…

For every $d\in\mathbb Z_{>0}$ and every positive divisor $k$ of $d$, let $\Omega_{d,k}^{\mathbb P^2/E}(\hbar)\in\mathbb Q[\![\hbar]\!]$ be the higher-genus relative BPS invariant…

Let $Q$ be the quiver associated with a noncommutative crepant resolution $A=\mathbb{C}\{Q,W\}$ of the threefold setup, let $C$ be the flopping curve, and let $\Phi(0,1)$ be its di…

Let $\beta$ be a curve class, let $(S,E)$ be general with respect to $\beta$, and let $P,P'\in E(\beta)$ be $\beta$-primitive points. Primitive-point independence conjecture. … Thi…

Assume that $S$ is a del Pezzo surface and that $\mathcal{M}_\beta$ is the moduli space of one-dimensional stable sheaves on $S$ with class $\beta$ and holomorphic Euler characteri…

Assume that $S$ is a del Pezzo surface and $E$ is an anticanonical divisor. Let $P\in E(\beta)$ be $\beta$-primitive, where $m^P_\beta$ is the log BPS number at $P$ and $n_\beta$ i…

Let $(S,E)$ be a rational log Calabi-Yau surface with smooth divisor and let $\beta\in\operatorname{H}_2(S,\mathbb{Z})$. For $P\in E(\beta)$, let $m^P_\beta$ denote the log BPS num…

Ooguri–Vafa integrality conjecture. One has

Index-factorization conjecture. The index satisfies

Categorification conjecture. There are series $\widehat Z_b(q)$, convergent for $|q|<1$, such that

Relative BPS integrality conjecture. For every $d\geq 1$,

Takahashi's conjecture. For all $d\geq 1$,

Let $r_{g,h}$ denote the BPS count associated to a class $\beta$ on a K3 surface whose norm square satisfies … Let $q$ and $y$ be formal variables. Katz–Klemm–Vafa product formula.…

Let $E\subset\mathbb P^{2}$ be an elliptic curve with a chosen group structure whose zero element is a flex point, and let $P\in E$ be a point of order $3d$. Let $m_{d}$ be the num…

Equivariant Gopakumar–Vafa correspondence. The Gopakumar–Vafa formula holds:

Let ${\mathcal K}$ be a knot, let ${\mathcal G}_R({\mathcal K})$ denote its Kauffman invariant in representation $R$, and let ${\mathcal H}_{(R_1,R_2)}({\mathcal K})$ denote the HO…