10 problems

An extremal transition is a process connecting topologically distinct Calabi–Yau manifolds through a singular Calabi–Yau variety and its smoothing or resolution. Candelas–de la Oss…

Let $L$ be an extremal transition and let $L_m$ be a proposed mirror surgery. Suppose $(X,Y)$ is a mirror pair and performing $L$ on $X$ produces $\widetilde{X}$. Mirror surgery co…

Let a flop be the birational surgery described for Calabi–Yau 3-folds, and let an extremal transition be an extremal contraction followed by smoothing, or the reverse surgery. Modi…

A Calabi–Yau threefold $X$ is a compact Kähler threefold with $c_1=0$, equivalently with trivial canonical line bundle $K_X$. Its complex and Kähler structures have a moduli space…

Extremal-transition conjecture. There exist a compactification $\overline{\mathcal{M}}$ containing $P$, boundary components $\Delta_{k+1},\dots,\Delta_r$ along which $X$ acquires c…

Let $X$ be a Calabi–Yau threefold, and let a type III primitive contraction contract a divisor $E$ to a curve $C$. A transition arising from this contraction is obtained by followi…

Let $X$ and $Y$ be smooth projective varieties related by a primitive extremal transition. Denote by $\mathcal{H}(X)$ the ambient part of the quantum $D$-module of $X$. After analy…

Let $X_L$ and $X_R$ be asymptotically cylindrical Calabi--Yau threefolds used to construct a twisted connected sum $G_2$-manifold by orthogonal gluing, and let an extremal transiti…

Smoothness conjecture. If a gauged linear $\sigma$-model is defined by pointsets ${\Scr A}$ and ${\Scr B}$ which are both ${\Scr A}$-complete and ${\Scr B}$-complete, then this mod…

Extremal transition conjecture. The following hold: (1) the structure constants of the small quantum product of $X_{\rm res}$ are polynomials in $q_1,\dots,q_l$ with coefficients r…