Title: Cobordism Categories and Spaces of Manifolds.
Abstract: Cobordisms have been one of the central objects in topology since the pioneering work of Rene Thom, which provided the first link between manifolds and homotopy theory. In more recent years, there has been much focus on cobordism categories. These play a fundamental role in the study of topological quantum field theories and automorphism groups of manifolds. A fundamental result in this field is the theorem of Galatius, Madsen, Tillmann, and Weiss, which describes the homotopy type of the classifying spaces of smooth cobordism categories. Galatius and Randal-Williams later simplified the proof of this result with the use of spaces of manifolds and scanning techniques. Besides giving an overview of this field of research, I will discuss the analogs of the theorem of Galatius, Madsen, Tillmann, and Weiss for topological and PL manifolds. The topological case is joint work with Alexander Kupers.