The theory of algebraic transformation groups, i.e. actions of algebraic groups on algebraic varieties, is one of the classical areas of algebra and algebraic geometry. It has many interconnections with combinatorics, differential geometry, algebraic group theory, Lie groups, Lie algebras, and representation theory.
One of the main objects of our study is affine algebraic varieties. On the one hand, affine geometry studies local properties of arbitrary algebraic varieties, on the other hand, it gives a geometric interpretation of natural questions in commutative and differential algebra. The combination of algebraic, geometric, representation-theoretical and combinatorial methods allows us to use a wide range of tools in modern mathematics and obtain original results.
The laboratory organizes regularly conferences, schools and seminars in affine algebraic geometry and transformation groups. Within the laboratory, students have an opportunity to work on modern research projects, to publish their results in leading mathematical journals and to take part in international collaborations.
- Automorphism groups of algebraic varieties
- Varieties with torus actions and graded algebras
- Additive actions on complete varieties
- Locally nilpotent derivations
- Cox rings and their applications