Jeffrey Danciger: Affine geometry and the Auslander Conjecture

Mathematics Colloquium at Rutgers University - Newark.

March 30, 2016

Speaker: Jeffrey Danciger (University of Texas at Austin)

Title: Affine geometry and the Auslander Conjecture

Abstract: 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 sketch a new construction of non-compact complete affine manifolds with non-solvable fundamental group whose cohomological dimension is arbitrarily large. At the heart of the issue is understanding the dynamics of discrete groups acting by affine transformations of R^n.

[Suav]

Jeffrey Danciger, thank you sir, and in this way thanks to all the people who show us how it is to climb in the first position and all those who made it possible for me to understand at least a bit of what you say. A bit like a "fog of war" experience.