9 problems

Let $(X,\Phi)$ be a $\operatorname{Spin}(7)$-manifold, and let a toric Cayley mean a Cayley four-fold of the toric type intended in the source. The Spin(7) SYZ fibration conjecture…

Let $(X_t,\omega_{CY,t})$ be a polarised degeneration family of $n$-dimensional Calabi–Yau manifolds near the large complex structure limit, and let $\mu_t$ denote the normalised C…

Metric SYZ conjecture. For any $0<\varepsilon\ll1$, there exists $\delta>0$ such that if $|t|<\delta$, then there exists a special Lagrangian $\mathbb{T}^d$-fibration on an open su…

Let $\omega_i$ be the sequence of symplectic forms considered in the paper, let $B_0$ be the base of the special Lagrangian fibration, and let $\mathcal{U}_i\subseteq B_0$ be a cha…

Let $Y$ be a toric Fano surface, and let $W(u)$ denote the superpotential of a special Lagrangian torus fibre $L_u$; let $W^{HV}$ be the Hori–Vafa superpotential. Special Lagrangia…

Combinatorial discriminant conjecture. After choosing suitable triangulation data of the dual polytope, the image of each component of $C^{(i)}$ retracts to a codimension-one subse…

Gross–Ruan topological conjecture. The discriminant and monodromy data of the fibration satisfy properties (i)--(iv) above, including the specified positive and negative vertex mod…

Let $X$ and $\check{X}$ be Calabi–Yau manifolds which are mirror to each other. A special Lagrangian torus fibration is a fibration with torus fibers calibrated by the special Lagr…

Let $X$ be a Kähler manifold, let $D\subset X$ be a smooth Calabi–Yau hypersurface in the anticanonical linear system, and let $\pi:X\setminus D\to B$ be a reasonable special Lagra…