5 problems

Let $\varphi$ be a closed $G_2$-structure, meaning a positive closed three-form, on a closed $7$-manifold. Consider the Laplacian flow … A closed $G_2$-structure has sufficiently s…

Compactness conjecture. Let $\{\phi_n\}$ be a sequence of volume-one $G_2$ forms in a connected component of $\mathfrak{M}_1$. If

Embedding conjecture. The chiral de Rham complex of $M$ admits an embedding of the SUSY vertex algebra $\mathrm{SV}_a$, where the value of $a$ is determined by the eigenvalue…

Lê's conjecture. If the assumption that $M$ is closed is removed, then $M$ admits a $\widetilde{\mathrm{G}}_2$-structure if and only if it is spin.

Embedding-reduction conjecture. The embedding problem for hypo $SU(2)$-structures and $SU(3)$-structures might be reduced to the embedding problem for $G_{2}$-structures.