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These are lecture notes prepared for the summer school "Geometric, algebraicand topological methods in quantum field theory", held in Villa de Leyva inJuly 2017. Our goal is to provide an introduction to a conjecture of Chern thatstates that the Euler characteristic of a closed affine manifold vanishes. Wepresent part of the history and motivation for the conjecture as well as somerecent developments. All comments and corrections are most welcome!

Lectures on the Euler characteristic of affine manifolds