6 problems

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…

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 $(\psi_t)_{t\in(-T,T)}$ 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…

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 $3$…