4 problems

Let $R={\mathbb Z}[t]$ or $R={\mathbb Z}[t,t^{-1}]$, let $K(R)$ be the kernel of the evaluation map $SL_2(R)\to SL_2({\mathbb Z})$, and let ${\mathcal W}$ be the kernel of the indu…

Let $X$ be a projective variety, and let $\rho$ be a reductive representation of its fundamental group, assumed reductive after restriction to the fundamental group of the normaliz…

Let $X$ be a variety, let $X_B$ be the $n$-groupoid truncation of the homotopy type of $X^{\rm top}$, and fix a representation $\rho$. The Betti relative Malcev completion is…

Hain's conjecture. The relative completion $\mathscr G$ should carry a nonabelian mixed Hodge structure in general. This structure should be a quotient of…