8 problems

Non-Abelian gauge-fixing conjecture. There exists a gauge transformation $\lambda\in W^{1,p}(\operatorname{Ad}P)$ such that

Realisability conjecture. The bundles $Q$ and $Q'$ are equivalent if and only if $\varphi$ is realisable.

Associated-connection conjecture. The connection $\eta$ can be seen as an associated connection to a principal connection on $(\hat{P}, M, \pi_{\hat{P}, M}, \operatorname{Aut}(G))$…

Rational cohomology conjecture. The rational cohomology of $\mathscr{B}un_{G,X}^0$ is

Let $\Gamma$ be a finite group and $(E,X,\pi)$ be a principal $\Gamma$-bundle, where $(X,\widetilde{\mathbb{Z}}_X)$ is a ringed, connected, compact, Hausdorff, and PCVP space. Supp…

Classifying-map conjecture. Under the relevant zigzag of equivalences, the map $N\alpha_*$ is a classifying map for $p:E\to B$: its homotopy class coincides with…

Let $R$ be a regular local ring, let $K$ be its field of fractions, and set $U:=\operatorname{Spec} R$. Let $\mathbf G$ be a reductive group scheme over $U$, and let $\mathcal G$ b…

Let $C$ be a smooth projective curve and let $G$ be a reductive group scheme over $C$. Let $P\subset G$ be the canonical parabolic subgroup. Behrend's conjecture. One has … This is…