Title: The algebraic geometry of the Putman-Wieland conjecture Abstract: Suppose we are given an unramified covering $\Sigma_{g'} \to \Sigma_g$ of topological surfaces with $g \geq 3$. The Putman-Wieland conjecture predicts that the action of the mapping class group of $Mod_{g,1}$ on $H_1(\Sigma_{g'})$ has no nonzero fixed vectors. More generally, one might predict that this action has big monodromy in a suitable sense. The Putman-Wieland conjecture is closely related to Ivanov's conjecture, predicting that finite index subgroups of the mapping class group (for $g \geq 3$) have finite abelianization. We will discuss joint work with Daniel Litt and Will Sawin making progress toward the Putman-Wieland conjecture, as well as several big monodromy variants. The methods are Hodge theoretic in nature via studying the derivative of an associated period map.