Étale Fundamental Groups -- a geometric and topological approach to fundamental groups in algebraic geometry

Abstract: This thesis explores the notion of fundamental groups across three mathematical settings. We begin with the classical topological theory of covering spaces, highlighting its structural analogy with Galois theory. We then follow Grothendieck in transporting these ideas to algebraic geometry. The inadequacy of the Zariski topology motivates the étale topology, from which the étale fundamental group is constructed and compared to its topological counterpart via transcendental methods. Finally, we linearise the theory through Tannakian duality, where fundamental groups are recovered as automorphism groups of fibre functors on certain monoidal categories, a framework broad enough to encompass étale, topological, and motivic Galois groups alike.