Departmental Colloquium
Jeffrey Danciger
UT Austin
Affine geometry and the Auslander Conjecture
Abstract: [The speaker will be part of the Big ideas in Dynamics and Geometry workshop.]
The Auslander Conjecture is an analogue of Bieberbach’s theory of Euclidean crystallographic groups in the setting of affine geometry. It predicts that a complete affine manifold (a manifold equipped with a complete torsion-free flat affine connection) which is compact must have virtually solvable fundamental group. The conjecture is known up to dimension six, but is known to fail if the compactness assumption is removed, even in low dimensions. We discuss some history of this conjecture, give some basic examples, and then survey some recent advances in the study of non-compact complete affine manifolds with non-solvable fundamental group.
Tools from the deformation theory of pseudo-Riemannian hyperbolic manifolds and also from higher Teichm\"uller theory will enter the picture.
Friday September 27, 2024 at 3:00 PM in 636 SEO