9 problems

Let $\mathcal M_k^0$ be the reduced moduli space of $\operatorname{SU}(2)$ monopoles of magnetic charge $k$ on $\mathbb R^3$, and let $\widetilde{\mathcal M}_k^0$ be its universal…

Let $\mu_0\in\mathfrak g$ be nonzero, let $\mu:[0,\delta]\to\mathfrak g$ be a regular deformation of $\mu_0$, meaning that $\mu(t)$ is regular for all $t>0$, and set…

Let ${\mathcal M}(u,\mu,k)$ be the framed moduli space of monopoles of type $(\mu,k)$, with its natural $L^2$ metric. HyperKähler conjecture. The spaces ${\mathcal M}(u,\mu,k)$ are…

Let ${\mathcal M}$ be the monopole moduli space, let ${\mathcal M}^{\infty}$ be the moduli space of boundary data at infinity, and let … be the boundary map. For monopoles…

Smooth-structure conjecture. Natural smooth structures should exist on these spaces such that these bijections are diffeomorphisms. Establishing this would give a smooth geometric…

Murray–Singer's conjecture. The space ${\mathcal M}(u,\mu,k)$ has dimension divisible by $4$, and its natural $L^2$ metric makes it into a hyperKähler manifold. This refines the cl…

Let $G$) be a compact Lie group and let the framed monopole moduli space be stratified by the possible asymptotic values of the monopole charge. Each stratum is a union of submanif…

The paper considers a newly derived eight-dimensional hyper-Kähler metric and the known eight-dimensional generalizations of the Taub–NUT metric associated with the rank-two simple…

The paper constructs an eight-dimensional noncompact hyper-Kähler metric, denoted here by $g$, and refers to a known metric with a discrete $Z_2$ isometry that may be obtained from…