Lectures on the Euler characteristic of affine manifolds

TL;DR

This paper introduces the conjecture that the Euler characteristic of closed affine manifolds is zero, discussing its historical context, motivation, and recent progress in understanding this geometric-topological relationship.

Contribution

It provides an accessible overview of Chern's conjecture, including historical background and recent developments, aimed at advancing understanding in affine geometry and topology.

Findings

Discussion of the conjecture's motivation and history

Summary of recent progress and partial results

Open questions and directions for future research

Abstract

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