28 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…

Universal-cover conjecture. The universal covering space $\widetilde{M}$ is isometric to $G/\operatorname{\textsl{SU}}(2)$ equipped with its unique $G_2$ structure with parallel to…

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 $M$ be a $G_{2}$-manifold, and let $C$ and $C^{\prime}$ be nearby coassociative submanifolds of $M$. Consider instantons in $M$ with small volume and boundary lying on…

Fiber-wise duality conjecture. This fiber-wise duality should work remarkably well when there is some residual supersymmetry and an underlying adiabaticity.

Let $B$ be a compact $3$-manifold, let $\Sigma$ be a smooth link in $B$, and let $h$ be a branched maximal section of the affine bundle $E_{\chi}$ associated with a representation…

In a $G_2$ manifold equipped with a Kovalev–Lefschetz fibration, gradient cycles are cycles in the base playing the role of tropical curves for the enumeration of associative subma…

A compact holonomy $\mathrm{G}_2$ manifold is a compact $7$-dimensional manifold admitting a metric whose holonomy is exactly $\mathrm{G}_2$. The known topological constraints incl…

$G_2$-Goldberg conjecture. An Einstein closed $G_2$-structure $\varphi$ must be torsion-free; in particular, $g_{\varphi}$ is Ricci flat.

Let $(\overline{X},\varphi_{CS})$ be a nearly $G_2$ conifold with a singularity at $x_0$ modelled on the $G_2$ cone $(C=C(L),\varphi_C)$. An obstruction conjecture for nearly $G_2$…

Let $f$ be the fibration defining the $G_2$-spaces $M_\theta:=f^{-1}(\theta)$, and let the spectral cover associated to $\theta$ be given. The spectral-cover conjecture. The singul…

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 $(Y,\varphi,g)$ be a compact $7$-manifold with holonomy $G_2$, and let a flag structure on $Y$ be the structure defined by assigning signs to flagged submanifolds and satisfyin…

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 $(Y,\psi_0)$…

Let $M$ be the underlying manifold, and let $(\varepsilon_i,\Psi_i,{\bm\xi}_i,A_i)$ be a sequence of solutions of the blown-up ADHM$_{1,k}$ Seiberg--Witten equation satisfying … wi…

Donaldson's conjecture. There exists a deformation of $\underline{\omega}$ through cohomologous hypersymplectic structures to a hyperkähler triple. This conjecture asserts that, up…

Combined-count deformation conjecture. A combination of the count of $G_2$-instantons and the count of multi-monopoles, summed over all associative submanifolds, has the correct de…

Let $(X,varphi)$ be a $G_2$-manifold equipped with a monopole gerbe $(\mathcal{G},F)$, and consider the analogue for gerbes of the condition that a connection on a complex line bun…

Let $(X^7,g)$ be a compact $G_2$-manifold, and let coassociative submanifolds be $4$-dimensional submanifolds calibrated by the $G_2$ $4$-form. Joyce's conjecture. It may be possib…

Let $K_f$ be a weighted Calabi–Yau link equipped with an $S^1$-invariant ${\rm G}_2$-structure, and let $\nu$ denote its Crowley–Nordström invariant. The invariant takes values in…

Adiabatic limit conjecture. If there is a positive section $h$ of ${\cal H}_{\chi}$ which avoids excess $-2$ classes, then for sufficiently large $R$ there is a closed positive $3$…

The half-flat structures are described by solution curves in the spaces $\mathcal H_0=\mathcal H_{(0,0)}$ and $\mathcal H_{(1,-1)}$, with associated $\mathrm{G}_2$-metrics and para…

Let $a,b$ be arbitrary parameters satisfying $ab\neq 0$, and let $a,b,c$ be arbitrary parameters satisfying $abc\neq 0$. No-new-solutions conjecture. The corresponding algebraic sy…

Positivity and trace bound conjecture. The tensor $T_{ab}$ remains positive along the flow, and $\operatorname{Tr}T$ satisfies $$ along the flow.