4 problems

Let $(M,a)$ be a closed, area-weighted topological map, embedded in a closed, connected, orientable surface, and let $G_N=\mathrm{U}(N)$. A loop is included in a disc if its drawin…

Let $\Sigma$ be a closed, connected, orientable, two dimensional, Riemannian manifold, the Euclidean plane $\mathbb{R}^2$, or a disc in $\mathbb{R}^2$. Let $G_N$ be a classical gro…

Let $\Sigma$, $U$, $C$, $a$, and $c=\operatorname{area}(\Sigma)-a$ be as in the preceding conjecture, with $C$ a simple closed curve in the topological disk $U$. For…

Let $\Sigma$ be a surface of area $\operatorname{area}(\Sigma)$, let $U\subset\Sigma$ be a topological disk, and let $C$ be a simple closed curve in $U$. Assign area $a$ to the int…