512 problems

Let $X$ be a smooth projective variety with an algebraic action of a complex torus $T_{\mathbb C}$, and let $Y=X/\!/T_{\mathbb C}$ be a smooth GIT quotient with no orbifold singula…

Gopakumar–Vafa integrality conjecture. The BPS invariants $n_{h,\beta}(X)$ defined by this equation should be integers for all $h\geq 0$ and $\beta\neq 0$.

Virasoro conjecture. For any smooth projective variety,

Let $(X,D)$ be a log Calabi–Yau surface pair with $D=D_1+\cdots+D_l$, and define … where $\mu$ is the Möbius function. Klemm–Pandharipande integrality conjecture. When $l=2$, the q…

Gromov–Witten/Donaldson–Thomas correspondence. The GW theory and the DT theory of $X$ should be equivalent after a change of variables.

Let $\mathcal{X}$ be a hard-Lefschetz Gorenstein orbifold and let $Y$ be its crepant resolution. Let the GW potentials encode the Gromov–Witten theories of $\mathcal{X}$ and $Y$. B…

Let $X=\mathcal{O}_{\mathbb{P}^1}(-1)\oplus\mathcal{O}_{\mathbb{P}^1}(-1)$ be the resolved conifold equipped with the anti-diagonal $C^*$-action on the fibers, and let the generati…

The paper considers cohomological Donaldson–Thomas theory as a limit of the K-theoretic computations developed there. Donaldson–Thomas/Gromov–Witten correspondence. Cohomological D…

Pandharipande's Gopakumar--Vafa integrality conjecture. The primary Gromov--Witten theory of threefolds is governed by certain integers, called Gopakumar--Vafa invariants or BPS in…

Virasoro constraints conjecture. For all $k\geq -1$ and $D\in\mathbb D_{\mathrm{PT}}^X$,

Let $M_N^k$ be a degree-$k$ hypersurface in ${\mathbb P}^{N-1}$, let $w({\cal O}_{h^a}{\cal O}_{h^b}|\prod_{p=0}^{N-2}({\cal O}_{h^p})^{m_p})_{0,d}$ denote its genus-$0$ multi-poin…

Crepant resolution conjecture. The transformation $\mathds{T}$ satisfies

Crepant transformation conjecture. Varieties related by crepant transformations have equivalent Gromov–Witten theories.

Let $(\mathbf{Q},\mathbf{W},\mathbf{v})$ be a quiver with potential and dimension vector, with associated gauge group $\mathsf{G}$, representation space $\mathsf{N}$, stability con…

The equivariant Gromov–Witten theory of $\mathbb P^1$ is governed by the equivariant Toda hierarchy. Toda conjecture. This is an analogue of the Witten conjecture connecting an enu…

Multiple cover formula.

Let $X$ be a nonsingular projective $3$-fold. The Gromov–Witten theory of $X$ consists of its Gromov–Witten invariants, and the Donaldson–Thomas theory consists of its Donaldson–Th…

The remodeling conjecture concerns a toric Calabi–Yau threefold and its mirror curve, on which the Eynard–Orantin topological recursion is defined. Remodeling conjecture. The open…

Holomorphic anomaly equations. Assuming the quasi-Jacobi form property,

A compact symplectic manifold or orbifold has symplectic rational connectedness, understood as the symplectic generalization of rational connectedness for projective varieties. Sym…

Let the spectral curve be … Here the stationary Gromov–Witten theory of ${\mathbb P}^1$ refers to Gromov–Witten invariants with stationary insertions, and topological recursion is…

Let $M$ be a projective manifold whose genus $0$ Gromov–Witten potential is semisimple, and let ${\mathscr D}_{Giv}$ be the formal total potential constructed from it by Givental's…

Eguchi–Hori–Xiong conjecture. The generating function satisfies the Virasoro conditions

The $r$-spin theory is a cohomological field theory whose correlators are intersection numbers on the moduli space of $r$-spin curves. Its big phase space potential function is def…

Two-partition Hodge-integral formula. One has