초록: Let X be a homogeneous space for a Lie group G. A (G,X)-structure on a manifold M is an atlas of coordinate charts valued in X, such that the changes of coordinates locally lie in G. It is a fundamental question to ask how many ways one can put a (G,X)-structure on M, i.e. what is the space of (G,X)-structures on the manifold M? In this talk, I will explain the strong interaction between the space of (G,X)-structures on M and the space of representations of the fundamental group of M into G. In particular, I will describe the current understanding of these spaces, focusing on the case when X is real projective space and G is the group of projective automorphisms of X.
