The Chabauty-Kim method is a technique for studying rational and integral points on curves using ideas inspired by Grothendieck's anabelian programme and Deligne's motivic philosophy on fundamental groupoids.

Strikingly, despite the highly abstract theory which underpins the method, Chabauty-Kim is actually a practical method for computing rational points on curves, and has been used to determine rational points on many different modular curves for which other computational methods do not succeed.

In this five-lecture course, we will give an overview of the Chabauty-Kim method, from theory to practice. Starting from Coleman's explicit Chabauty method, we will explain Kim's key insight, that Chabauty-Coleman admits a "non-abelian lift" which can be described in terms of motivic and especially étale fundamental groups. We will explain how Kim's general method allows us to identify certain iterated Coleman integrals which vanish on the rational points of curves, and finish by outlining how the theory works in explicit computations.

Keywords: Chabauty-Kim method, arithmetic fundamental groups, mixed motives, Coleman integration, rational points.

Lecturers: A. Betts (Cornell, US), D. Corwin (Ben Gurion, IL), M. Lüdtke (Oldenburg, DE - Ben Gurion, IL)

※ This event is part of the special year ``Arithmetic, Homotopy, and Geometry 2027-28''.