Benjamin Bakker of the Institute for Advanced Study presents a Members' Colloquium on the fundamental group of a complex algebraic variety, an invariant that turns out to be tightly linked to the variety's geometry. He frames the talk around two questions: how many simply connected algebraic subvarieties a variety can contain, and how many global holomorphic functions its universal cover admits. The second question connects to Shafarevich's conjecture, which predicts that for smooth projective varieties the universal cover has enough holomorphic functions to separate points up to compact ambiguity. Bakker describes joint work with Yohan Brunebarbe and Jacob Tsimerman that fully answers both questions when the fundamental group has a faithful linear representation, drawing on techniques from non-abelian Hodge theory. The talk runs 67 minutes and assumes graduate-level familiarity with algebraic geometry and topology.