25 problems

Let $(A_i,$Phii$)$ be Calabi–Yau monopoles with increasingly large mass parameters on a Calabi–Yau 3-fold, and let $L$ be a special Lagrangian 3-fold near which their curvature con…

Let $X$ be a Calabi–Yau threefold, and consider type IIA string theory compactified on $X$. Fix the magnetic D6- and D4-brane charges $P_{6}$ and $P_{4}$, and sum over electric D2-…

Joyce's transversality conjecture. This transversality condition holds in any almost Calabi–Yau manifold whose Kähler class is sufficiently generic.

Consider a generic almost Calabi--Yau $3$-fold and a $\mathrm{U}(1)$-invariant local model for codimension-two singularities in a special Lagrangian fibration, in which two codimen…

Suppose $M$ and $\hat M$ are mirror Calabi--Yau $3$-folds. A special Lagrangian torus fibration is a fibration whose generic fibers are special Lagrangian $3$-tori. The SYZ conject…

Let $X$ be a smooth resolution preserving the Calabi–Yau condition of the cone $X_\circ=C(L)$ over a $(2d-1)$-dimensional toric Sasakian manifold $L$, where $X_\circ$ is a non-comp…

Thomas–Yau conjecture. If the flow has a finite-time singularity, then $L$ can be decomposed into a graded connected sum $M_1\# M_2$ satisfying

Let $(\overline{X},\omega_{CS},\rho_{CS})$ be a nearly Kähler conifold with a singularity at $x_0$ modelled on the Calabi--Yau cone $(C=C(L),\omega_C,\rho_C)$. An obstruction conje…

Let the central Hirzebruch scroll be the limiting member of the deformation family of Hirzebruch scrolls described above, and let its Calabi–Yau hypersurface be Tyurin-degenerate,…

PT-DT correspondence. The Donaldson–Thomas and Pandharipande–Thomas generating functions are equal up to a factor of MacMahon's generating function for plane partitions.

Let $r^{\prime}:H^{\operatorname{LG}}\to H^{\operatorname{CY}}$ be the correspondence map, and let … and … be the Landau–Ginzburg and Calabi–Yau $tt^*$ structures. Conjectural…

Let the big Landau–Ginzburg B-model be the full Landau–Ginzburg model considered in the paper, and suppose that the usual Calabi–Yau B-model is too small to correspond to it. Conje…

Let $M_{mar}$ be the marginal-deformation subspace of the deformation space $M$. Let…

Let $f=z_0^5+\cdots+z_4^5$ define the quintic Calabi–Yau hypersurface $X_f\subset\mathbb{CP}^4$, let $G$ be an admissible subgroup satisfying $\langle J\rangle<G<G_{\max}$, and set…

Strong Strominger–Yau–Zaslow conjecture. For $|t|\ll 1$, the 3-fold $X_t$ admits a special Lagrangian $T^3$-fibration over a base $B$ homeomorphic to a 3-sphere. The base $B$ is eq…

Grassmannian Calabi–Yau correlator conjecture. The genus-0 1-point A-model correlator has the expansion

Let ${\mathcal M}_{\rm par}$ be the moduli space of singular parabolic Higgs bundles with $n$ simple poles in their connection, and let ${\mathcal M}_{\rm wild}$ be the moduli spac…

Let a compact singular elliptically fibered Calabi–Yau $3$-fold have singular fibers associated to ${\mathcal G}$-symmetry in codimension $1$ over the base, admit a crepant resolut…

Let $X$ be a local Calabi–Yau manifold with a $\mathbb C^*$ isometry, let $g_X$ be the relevant integer, let $\mathbf C$ be the matrix appearing in the blowup equation, and let…

Let $X\in\widetilde{\mathscr{KS}}(n,\kappa)$ be a Calabi–Yau conifold and let $x_0\in X$. The tangent space at $x_0$ is the limiting tangent object associated with the conifold str…

Conjectural genus-one formula. The displayed identities give the genus-one Gromov–Witten generating series for $X$, with the odd- and even-dimensional cases as specified.

Metric contraction conjecture. The singular AC metrics on the $r$-th root of $K_{\mathbb{P}(E)}$ contract along one-parameter families to the smooth AC metrics on $E\otimes L$ by s…

Let $D\geq 1$ and consider the $2D+3$-dimensional case. Let $q=e^t$, let $a$ be the insertion index, and let $n_{2d-1}$ denote the disk invariant of degree $2d-1$. Modified multipl…

A Calabi–Yau space is a quadruple $(M,J,g,\Omega)$, where $(M,J)$ is a complex compact $n$-dimensional manifold. Berger graphs are constructed from reflexive weight vectors and New…

Kähler-cone conjecture. The moduli space ${\mathcal M}$ should itself be a Kähler cone, with $\rho_0$ playing the role of the radial coordinate.