4 problems

Let $X$ be a projective manifold, let $Z$ be a positive-dimensional subvariety, let $Z_{\operatorname{norm}}$ be its normalization, and let…

Let $S_g$ be a closed orientable surface of genus $g$, and let $\rho$ be the family of linear representations of the mapping class group $\operatorname{Mod}(S_g)$ arising from Kont…

Let $M$ be a compact $3$-manifold and let $\gamma\in\pi_1(M)\setminus\{1\}$. A group is residually $\operatorname{PGL}_2$-finite if, for every nontrivial element, there is a finite…

Let $\BN$ denote the braid group on $N$ strands, and let a faithful representation mean an injective homomorphism from $\BN$ into a linear group of dimension smaller than that of t…