Abstract: This talk will give a historical motivation for stacks from the nonexistence of moduli spaces, give a non-technical definition of a stack, and describe some examples.

Abstract:

Abstract: Let $G$ be a reductive group. A $G\times G$-variety $X$ is called an equivariant compactification of $G$ if $X$ is normal, projective, and contains $G$ as an open and dense orbit. Regular compactifications and reductive embeddings are the main source of examples. In the first case, the equivariant cohomology ring has been explicitely described by Bifet, de Concini, Procesi and Brion. Loosely speaking, it depends mostly on the torus embedding part and the structure of the $G\times G$-orbits. As for the second class, Renner has found that they have a canonical cell decomposition based on underlying monoid data. My goal in this talk is to give an overview of the theory of group embeddings, putting more emphasis on the monoid approach, and to describe the structure of the so called rational cells. Finally, I will explain how such cellular decompositions could lead to a further application of GKM theory to the study of reductive embeddings.