Morita theory and invertibility in bicategories

We present a bicategorical perspective on derived Morita theory for rings, DG algebras, and spectra. This perspective draws a connection between Morita theory and the bicategorical Yoneda Lemma, yielding a conceptual unification of Morita theory in derived and bicategorical contexts. This is motivated by study of Rickard's theorem for derived equivalences of rings and of Morita theory for ring spectra. Along the way, we gain an understanding of the barriers to Morita theory for DG algebras and give a conceptual explanation for the counterexample of Dugger and Shipley.;Additionally, we use a bicategorical context to develop and study general invertibility. Aside from introduction of basic definitions and foundations, the focus is a characterization of generalized Azumaya objects. We give tilting theory as an application of this characterization. A broader goal of this work is to help give a calculational foothold on Picard and Brauer groups;Chapter 5, Hinfinity Orientations on BP, is joint with Justin Noel. We show, at the primes 2 and 3, that no map from MU to BP defining a universal p-typical formal group law on BP is H infinity. In particular, no such map is Einfinity .;This builds on McClure's work on determining if Quillen's orientation on BP is an H2infinity map. By direct computation, we show that the necessary condition he derives for Quillen's orientation to be H2infinity fails at the primes 2 and 3. We go on to show that this implies the more general result above.;We also provide a reinterpretation of McClure's conditions in the language of formal group laws.