Title: Decompositions of 3-dimensional Alexandrov spaces
Abstract: Alexandrov spaces (with curvature bounded below) are
metric generalizations of complete Riemannian manifolds with a uniform
lower sectional curvature bound. Instances of Alexandrov spaces
include compact Riemannian orbifolds and orbit spaces of isometric
compact Lie group actions on compact Riemannian manifolds. In addition
to being objects of intrinsic interest, Alexandrov spaces play an
important role in Riemannian geometry, for example, in Perelman's
proof of the Poincaré Conjecture. In this talk, I will discuss the
topology and geometry of 3-dimensional Alexandrov spaces, focusing on
extensions of basic results in 3-manifold topology (such as the prime
decomposition theorem) to general three-dimensional Alexandrov
spaces. This is joint work with Luis Atzin Franco Reyna, José Carlos
Gómez-Larrañaga, Luis Guijarro, and Wolfgang Heil.