6 problems

Let $S$ be a curve of bidegree $(k,k)$ in ${\mathbb P}_1\times {\mathbb P}_1$. Let $L^r(a,b)$ denote the line bundle used in the hyperbolic-monopole spectral-curve construction, re…

The LeBrun metrics are generalized multi-centre metrics arising in the study of hyperbolic monopole motion. They describe geodesic motion in a simplified setting where one or more…

Nash's decomposition and symplectic-form conjecture. Nash's decomposition coincides with the tangent-space decomposition, and Nash's symplectic forms coincide with the forms…

Sutcliffe's rank conjecture. The conditions of $\mathcal{M}$ imply that the matrix $H$ has rank at least one. The conjecture concerns a nonvanishing condition needed in the constru…

Let $\mathcal{M}$ denote the conditions defining the monopole data, with matrices $M_3$ and $I_k$ as in the source. Sutcliffe's conjecture. In the $n=1$ case, the conditions of…

Let hyperbolic space have constant sectional curvature tending to zero, and let hyperbolic monopoles and Euclidean monopoles be the corresponding monopole solutions. Atiyah's conje…