Fundamental groups and rational points

One powerful tool in the study of rational solutions to equations comes from various theories of fundamental groups of varieties, most notably the etale fundamental group developed by Alexander Grothendieck. In this talk, we will outline how one can apply such algebro-topological tools to the determination of solutions to equations, both in terms of Grothendieck's famous Section Conjecture and (time permitting) the non-abelian Chabauty method of Minhyong Kim. There will be a focus on providing a general overview of the area, illustrated by examples, rather than setting out precise definitions, and we hope the talk should be widely accessible.