Global cohomology theories and applications

The study of cohomology theories defined on equivariant spaces has long been a key aspect of algebraic topology. In recent years it has become increasingly clear that many interesting cohomology theories defined on equivariant spaces admit the additional functoriality which characterizes a cohomology theory defined on topological stacks. Such cohomology theories are often referred to as global and prominent examples include Borel cohomology, complex K-theory and bordism. As often is the case in equivariant homotopy theory, we are more interested in "genuine" global cohomology theories which are in turn represented by global spectra. In this talk I will present conditions under which a global cohomology theory is represented by a global spectrum. As an application we will see that tempered and elliptic cohomology all admit global spectra. This talk is based on joint work with David Gepner and Sil Linskens.