24 problems

Let $X$ be a smooth projective threefold, let $\overline{\mathcal{M}}_{g,n}(X;\beta)$ be the moduli space of stable maps, and let $\tau_k(pt)$ denote the Gromov–Witten descendent i…

Let $X$ be a smooth projective variety, let $V$ be a fixed sheaf on $X$, and let $P$ be a moduli space of pairs consisting of a sheaf $F$ and a map $V\to F$, with universal pair…

Let $X$ be a smooth projective variety, let $M=M_\alpha$ be a moduli space of sheaves of topological type $\alpha$, and let $\mathbb{D}^X$ be the descendent algebra. Choose an alge…

Let $X$ be a simply-connected smooth projective 3-fold, let $\gamma\in H^{p,0}(X)$ with $p=2$ or $p=3$, and let $D\in\mathbb D_{\textup{PT}}^{X}$. For $k\geq 0$, the stable-pairs d…

DT/PT correspondence conjecture. For any such data,

Oblomkov–Okounkov–Pandharipande conjecture. If all insertions are stationary, meaning $\alpha_i\in H^{>0}(X)$, then the renormalized invariant is a rational function of $q$ regular…

DT/PT correspondence conjecture. For any $k_1,\dots,k_n$ and $\gamma_i\in H^*_{\mathrm{T}}(X)$,

Okounkov–Pandharipande's conjecture. The normalized degree-zero invariants

Equivariant DT descendent algebra conjecture. For $\gamma_i\in H^*_{\mathsf{T}}(X,\mathbb{C})$ and $\beta\in H_2(X,\mathbb{Z})$,

GW descendent polynomiality conjecture. The series

DT descendent polynomiality conjecture. The series

Stationary GW/PT correspondence conjecture.

Stable pairs descendent rationality conjecture. The series

Let $X$ be a nonsingular projective $3$-fold and let $\beta\in H_2(X,\mathbb Z)$ be a nonzero effective class. Let $d=\operatorname{div}(\beta)$ be the divisibility of the image of…

Let $X$ be a nonsingular projective $3$-fold, let $beta e 0$ be a class in $H2(X,mathbb{Z})$, and let $tau{ki}(gammai)$ be descendent insertions with $gammaiin H^(X)$. The descende…

Let $S$ be a $K3$ surface, let $\beta_2$ be the hyperplane-section class of a degree-two cover … branched along a nonsingular sextic $C_6\subset\mathbb{P}^2$, and let $\beta_5$ den…

Open Witten KdV conjecture. For every $n\geq 1$,

Relative descendent GW/Pairs correspondence. Under the variable change $-q=e^{iu}$,

Descendent GW/Pairs correspondence. Under the variable change $-q=e^{iu}$,

Relative descendent GW/Pairs correspondence.

Relative primary GW/Pairs correspondence.

Stable pairs descendent rationality conjecture. The partition function

Let $X$ be a nonsingular 3-fold, let $0\ne\beta\in H_2(X,\mathbb{Z})$, and let $\gamma_j\in H^*(X,\mathbb{Z})$ with nonnegative integers $i_j$. For the stable-pairs partition funct…

Stable-pairs rationality conjecture. The partition function $\mathsf Z_{\beta}^X\left(\prod_{j=1}^k\tau_{i_j}(\gamma_j)\right)$ is the Laurent expansion of a rational function in…