Algebraic transformation groups arise in many different contexts, e.g. as symmetry groups of certain classical objects or as automorphism groups in algebraic classification problems (moduli spaces and geometric invariant theory). Our research in this project is centered around the interplay between the geometric properties of a “space” on the one hand and the algebraic structure of its symmetries on the other. The symmetry group might help to get a deeper understanding of the structure of the space and to discover new interesting invariants. Typical general questions might be: How does the local structure of a space influence the global automorphisms? What geometric properties can be deduced from the existence (or the non-existence) of certain invariants?

For this proposal we have the following themes in mind.

• Galois correspondence for algebraic groups. There is a correspondence between reductive subgroups of GL(V) and certain subalgebras of T(V) ⊗ T(V*) where T (V) is the tensor algebra of V. This will lead to a unified approach to the classical First Fundamental Theorems and should help to discover several new one’s.

• Invariant Hilbert schemes and sheets. There is a strong connection between the sheets in a representation space V and equivariant Hilbert schemes in V . On one hand, the latter should help to understand the strange behavior of sheets in Lie algebras, and on the other, our knowledge of the geometry of representations of SL2 should provide us with new examples and a better insight into the corresponding invariant Hilbert schemes.

• Automorphisms of A³ and variables of k[x, y, z]. Recent results of Shestakov-Umirbaev about the automorphisms of A3 should help to get some new insight into the difficult question whether an embedded A² ⊂ A³ is always given by a variable.

• Unstable linear subspaces and polarizations. There is a very interesting connection between linear subspaces of the nullcone of a representation V of a reductive group G and the polarizations of the G-invariants of V which is not understood so far.

• Classification of wonderful varieties. The so-called spherical varieties are generalizations of the well-known toroidal embeddings and play a central and important roˆle in the theory of algebraic transformation groups. Their classification is not achieved yet, although quite a lot is known today. It became clear from recent work of several authors that the wonderful varieties will be a major object in this classification.