Low dimensional collapsing theory and soul theorem for Alexandrov spaces

In this talk we will give an overview of the collapsing theory for 3-dimensional manifolds with a lower sectional curvature bound. This theory first developed by Shioya and Yamaguchi served as the completion of the last claim in Perelman's Proof of Geometrization Conjecture. The basic techniques used in the study will be presented with emphasis on Perelman's Fibration Theorem and Soul Theorem. Collapsing theory with bounded sectional curvature will also be quickly reviewed so that we can see the differences/parallel structures between these two theory.