Dynamical methods in CR-geometry

The subject of Cauchy-Riemann Geometry (shortly: CR-geometry) was initiated in the classical work of Poincare and Cartan, and was further developed in later work of Tanaka, Chern and Moser, and others. CR-geometry is remarkable in that it lies on the border of several mathematical disciplines (Complex Analysis, Differential Geometry and Partial Differential Equations), and is an important tool for each of these areas.

In our joint research (partly joint with Rasul Shafikov), we have discovered a new face of CR-geometry. This is a “bridge” technique between CR-geometry and one of the fundamental areas of modern mathematics: Dynamical Systems. The technique allows for producing a certain vocabulary between the two theories.

We call the latter method the CR (Cauchy-Riemann manifolds) - DS (Dynamical Systems) technique. The CR -- DS technique has recently enabled us to solve a number of long-standing problems in CR-geometry, related to important degenerate structures appearing in CR-geometry.

The current research project is aimed to develop the CR -- DS technique in several directions, by employing modern methods in Dynamical Systems. We are going to actively collaborate with the Russian team, which consists of well known experts in both CR-geometry and Dynamical Systems. As a bi-product, we plan establishing certain new results in the area of Analytic Differential Equations.