10 problems

An associative submanifold is a $3$-dimensional submanifold calibrated by the $G_2$-structure. A singularity modeled on the cone over $T^2$ means that, locally, the singular associ…

The $\chi$-flow conjecture. The $\chi$-flow on $\Sigma\subset Q$ takes $\Sigma$ diffeomorphically onto the zero section.

Let $(X^4,\omega_1,\omega_2,\omega_3)$ be a hyperkähler $K3$ surface, and let $\alpha_1,\alpha_2,\alpha_3$ be irreducible $(-2)$-classes in $H^2(X^4;\mathbb{Z})$ satisfying…

Let $X$ be a K3 manifold, let $\alpha_1,\alpha_2,\alpha_3$ be $-2$ classes on $X$ satisfying … and let $H\subset H^2(X)$ be a maximal positive subspace corresponding to a hyperkähl…

A $\mathrm{G}_2$-structure on a $7$-manifold $M$ determines $\mathrm{G}_2$-instantons and associative $3$-folds, where the latter are semi-calibrated by the defining $3$-form. Dona…

Let $M$ be the compact $G_2$ manifold under consideration, and let the associatives $\tilde{C}_\gamma$ be the three-spheres obtained from $S^1_{x_3}$ and sections of the relevant e…

For every element $(\gamma,\hat\gamma)$ of $E_8 \oplus E_8$, let $\Sigma_{\gamma}^+$ and $\Sigma_{\hat\gamma}^-$ be three-chains on $Z_\pm$ with boundaries $e_1-e^1$, where…

Let $Y$ be a $G_2$-manifold, let $\psi$ denote its definite $4$-form, and let $\beta$ be the relevant homology class. Let ${\rm CMA}_\beta^\times(\psi)$ be the refined chain comple…

Let $(\psi_t)_{t\in(-T,T)}$ be a $1$-parameter family of closed, tamed, definite $4$-forms on $Y$. Let $\hat P$ be an unobstructed singular associative submanifold in $(Y,\psi_0)$…

Effective-associatives conjecture. The appropriate way to study such degenerations is to move to a wall in a “cone of effective associatives.”