Given a smooth manifold or orbifold M and a Lie group G acting transitively on a space X, we consider the space of all (G, X)-structures on M up to an appropriate equivalence relation. This space, known as the deformation space of (G, X)-structures on M, encodes information about how one can "deform" the (G, X)-manifold M. In this talk, I will provide a general definition of deformation spaces and character varieties, which capture the local structure of the deformation space. Additionally, I will introduce a class of orbifolds called the Coxeter orbifolds, for which deformation spaces can be computed using an approach due to the foundational work of E. Vinberg.

This talk aims to consider the attainability of the Hardy-type inequality in the bounded smooth domain with average-zero type constraint. Since the criteria of the attainability depends to the concentration-compactness type arguments, we will briefly introduce the results for some classical Hardy-type inequalities and the concentration-compactness arguments. Subsequently, we propose new function spaces that well define the new inequalities. Finally, we will discuss the attainability of the optimal constant of the inequality in the general smooth domain.