Fundamental Groups of Algebraic Varieties and the Shafarevich Conjecture

Explore the fundamental group π1(X) as an important invariant of complex algebraic varieties in this Members' Colloquium lecture from the Institute for Advanced Study. Delve into the topological nature of fundamental groups and their close connections to the geometry of algebraic structures on varieties X. Examine two elementary yet profound questions: determining how many simply-connected algebraic subvarieties a variety X contains, and understanding how many global holomorphic functions exist on the universal cover X̃. Learn about the famous Shafarevich conjecture, which asserts that for smooth projective varieties, the universal cover X̃ possesses sufficient global holomorphic functions to separate points up to compact ambiguity. Discover joint research findings with Brunebarbe and Tsimerman that provide complete answers to both fundamental questions when π1(X) admits a faithful linear representation, utilizing advanced techniques from non-abelian Hodge theory to bridge topology and algebraic geometry.