National Research University Higher School of Economics
Geometry of affine varieties and infinitely transitive actions
A finite-dimensional affine space is the natural habitat of many objects coming from domains as diverse as fundamental and applied mathematics, theoretical physics, and other mathematical sciences. It is therefore remarkable that the affine space itself and the family of its automorphisms give rise to a number of exciting open mathematical problems, such as the Jacobian Conjecture, the Cancellation problem, the Linearization problem, the Rectification problem (we still do not know how many lines are there in 3D (complex affine) space), or the problem of tame and wild authomorphisms. In the first part of this talk we will formulate these problems and briefly address the state of the art regarding their study.
In the second part we will present a new approach to the study of authomorphism groups. A group G is said to act on the set X in an infinitely transitive way if for any positive integer m each set of m different points in X can be mapped into any other such set by the action of an element of G. It was discovered recently that for many complex varieties the corresponding group of authomorphisms is infinitely transitive. We will show how this property is related to the flexibility of the variety and discuss the following surprising fact: under certain assumptions on the action, transitivity implies infinite transitivity.