11 problems

Let $\Phi_t$ be a one-parameter family of moduli-spaces arising from a one-parameter family of structures on the ambient manifold, where $\Phi_0$ compactifies to a fibration of tha…

A compact 7-manifold with holonomy $G_2$ is a Riemannian 7-manifold whose holonomy group is $G_2$. A coassociative submanifold is a four-dimensional calibrated submanifold in a…

Let $C=(S,\Theta,\xi)$ be a calibrated current of dimension $2$ in the setting where calibrated currents are pseudo-holomorphic, namely when the calibrating form is…

Let $C=(S,\Theta,\xi)$ be a $k$-dimensional calibrated current. If the calibrating form is $\Omega=\omega^l/l!$ with $k=2l$ on a symplectic manifold $(M,\omega)$ with compatible me…

Let $M$ be a $G_2$-manifold and consider calibrated fibrations on $M$, analogous to special Lagrangian or holomorphic fibrations on a Calabi–Yau threefold. Conjectural characteriza…

Let $M$ be an $8$-dimensional compact manifold admitting Calabi–Yau structures, let ${\cal M}$ be the moduli space of such structures, and let $E$ be a fixed $U(r)$-bundle over $M$…

Let $L$) be an asymptotically conical (AC) special Lagrangian $m$-fold in $\mathbb{C}^m$ with cone $C$, and let $k$ be the number of ends of $C$ at infinity. Write $b^1(L)$ for the…

A torsion-free ${\mathrm{Spin}}(7)$-manifold is a Riemannian manifold with torsion-free ${\mathrm{Spin}}(7)$-structure, and a smooth Cayley fibration is a smooth fibration whose fi…

Let $(M,\varphi)$ be a $G_2$-manifold and let $P\to M$ be a principal bundle with compact semisimple structure group. A $G_2$-instanton is a connection $A$ on $P$ satisfying … An a…

Coassociativity conjecture. The K3 fibres are coassociative four-cycles in $J$; equivalently, they are calibrated by $\star\Phi_3$.

The homology-class conjecture. Every homology class in