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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.

Determinantal Barlow surfaces and phantom categories I will report on the construction of an exceptional collection of maximal length 11 on the Barlow surface S. This can be used to show that in a small neighbourhood of S in the moduli space of determinantal Barlow surfaces, the generic surface has a semiorthogonal decomposition of its derived category into a length 11 exceptional sequence and a category with trivial Grothendieck group and Hochschild homology, called a phantom category. This is joint work with Christian B?hning, Hans-Christian Graf von Bothmer and Ludmil Katzarkov.