8 problems

Chern-class vanishing conjecture. All Chern classes of $\mathfrak{E}$ vanish.

Bruzzo and Graña Otero conjecture. The Higgs bundle $\mathfrak{E}$ is semistable with respect to some polarization $H$ and $\Delta(E)=0$.

Curve-semistability characterization. The following are equivalent:

Let $k$ be a field of positive characteristic, and let a Higgs bundle on a smooth projective variety over $k$ be nilpotent if its Higgs field is nilpotent, with its exponent define…

Let $X$ be a smooth complex projective variety, let $\mathcal O_X(1)$ be ample, and let $E$ be a slope-semistable locally free sheaf. Choose a finitely generated model…

Let $G$ be a 1-ended CAT(0) group, meaning a group acting geometrically on a proper CAT(0) space. Its end invariant $\operatorname*{pro}$-$\pi_{1}(\varepsilon(G))$ is defined throu…

Let $G$ be a 1-ended finitely presented group, and let $\varepsilon(G)$ denote its end. Write $\operatorname*{pro}$-$H_{1}(\varepsilon(G);\mathbb{Z})$ for the inverse system of fir…

Let $M$ be a closed aspherical manifold of dimension greater than $1$, and let $\widetilde{M}$ denote its universal cover. A space is strongly connected at infinity when its end ha…