Abstract: In contrast to the Riemannian setting, a Lorentzian manifold (M,g) is not known to possess any naturally induced distance function. I will first try to explain why that is, starting with some of the basics of spacetime (Lorentzian) geometry. We will then discuss a `null distance function' introduced in joint work with Christina Sormani, some of its properties, examples, and some open questions.
Abstract: To understand the geometry of nonpositively curved (NPC) spaces, it is natural to classify the various types of spaces that can occur. The Rank Rigidity Theorem for compact NPC manifolds separates the class of compact NPC manifolds into three very distinct types, and proves that nothing else can exist.
A version of Rank Rigidity has been conjectured for more general NPC spaces (CAT(0) spaces). In this talk, we discuss some progress toward this general conjecture, by reducing the problem to looking at patterns on
spheres. In particular, we prove the conjecture for certain NPC spaces with one-dimensional boundary. Unlike previous results in this area, there are no additional constraints on the CAT(0) space (such as a manifold or
Abstract: In this talk, I will try to explain the title. There will be 4-manifolds, many pictures, and very little background needed.
Abstract: In 1981, M. Gromov completed the proof that every manifold
admitting an expanding map is, up to finite cover, homeomorphic to a
nilmanifold. Since then it was an open question to give an algebraic
characterization of the nilmanifolds admitting an expanding map. During my
talk, I will start by introducing the basic notions of expanding maps and
nilmanifolds. Then I explain how the existence of such an expanding map
only depends on the covering Lie group and on the existence of certain
gradings on the corresponding Lie algebra. One of the applications is the
construction of a nilmanifold admitting an Anosov diffeomorphism but no
expanding map, which is the first example of this type.