16 problems

Let $G$ be a reductive group, let $C$ be a curve, and let $E$ be a principal $G$-bundle over $C$ with canonical parabolic reduction. Behrend's conjecture. The canonical parabolic r…

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.

Categorical Hopf symmetry conjecture. The categorical group $String(3)$ is equivalent to the categorical group of symmetries of the categorical Hopf map.

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 $G$ be a reductive algebraic group over $\operatorname{Spec}(\mathcal{O})$, and let $\mathfrak{X}_\Sigma$ be a toric scheme associated with a fan $\Sigma$. A toric principal…

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…

Let $P$ be a standard parabolic subgroup of $G$ with Levi component $M$. Let $\cal E$ be a $G$-bundle and $\cal E'_M$ be a semi-stable $M$-bundle with … where…

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…

A string datum consists of local data whose underlying bundle is a principal $\operatorname{Spin}(n)$-bundle $\bar{P}$. Gluing these local data is expected to produce an $S^1$-gerb…

Let $S$ be a smooth algebraic surface over a field $\Bbbk$, let $X\subset S$ be a smooth projective curve, and let $\widehat{S}$ be the formal completion of $S$ along $X$. For a si…

Parametrized classifying-space homeomorphism conjecture. The methods of Barratt and Heath adapt to give a homeomorphism

A principal connection is a gauge potential on a principal fibre bundle, and a deformation of the associated module of sections is the corresponding deformed module structure. Seib…

Let $G$ be a Lie group, let $H\subset G$ be a subgroup, and let $y\in\mathfrak{h}^{\perp}$, where the orthogonal complement is taken with respect to the relevant pairing on the Lie…

Let $G$ be a Lie group and let $P\to B$ be a $G$-principal bundle equipped with a fat connection. Suppose that its structure group reduces to a proper subgroup $H\subset G$; no fat…