In 1900, Hilbert asked (among many other things) if there are only finitely many n-manifolds locally isometrically modeled on Euclidean n-space. About 10 years later, Bieberbach proved that all such manifolds have virtually abelian fundamental group, and used this to solve Hilbert's problem. Inspired by this, we could study compact manifolds that have an affine structure (instead of a Euclidean structure) and ask what their fundamental groups can be. The Auslander conjecture states that the fundamental group of these manifolds are virtually solvable, and is still open. I will discuss the proof of this conjecture in dimension 2 (classical) and dimension 3 (Fried-Goldman).

Not conjectured by Auslander, but proven by him. However, asked by Milnor.
: Auslander's proof contains a flaw.
*: For complete affine manifolds, not necessarily compact ones. Noncompact counterexamples have been found by Margulis. Speaker(s): Wouter Van Limbeek (University of Michigan)