Classifying wonderful varieties
The 'Smoothness of the invariant Hilbert scheme of affine spherical varieties for the existence of wonderful varieties' (Smooth) project aimed at classifying a special class of spherical varieties, called wonderful, by proving Luna's conjecture. This theory states that there is a one-to-one correspondence between equally shaped classes of wonderful varieties and spherical systems. A spherical system is a coordinate system for three-dimensional space that is useful for analysing systems with some degree of symmetry about a point, such as within a sphere. Researchers proposed to provide the corresponding wonderful variety to a given spherical system by studying the geometric properties of the invariant Hilbert scheme, used for classifying problems of certain algebraic varieties. However, on commencement of this EU-funded project, partners discovered that the particular method of research was already in use by others at an advanced stage of development. They therefore decided to continue work on classifying spherical varieties via Luna's conjecture, but with different methods. Studies extensively analysing the combinatorics (i.e. measurements of structures in an algebraic context) of the so-called spherical systems resulted in a better understanding of their interplay with the geometry of wonderful varieties. This marked the beginning of the development of a more complete theory of wonderful varieties. Smooth researchers managed to solve technical problems that presented during their efforts to generalise Luna's original approach to the classification. Importantly, they also succeeded in formulating the strategy for fully proving Luna's conjecture, which until now has been only partially proven under certain hypotheses. Smooth project work offered a constructive approach to the classification, providing an algorithm to associate a wonderful variety to a given spherical system.