Regularity and Complexity in CR-geometry

Cauchy-Riemann Geometry (shortly: CR-geometry) is an area of mathematics going back to the research of H.Poincaré and E.Cartan, and lying on the border of several fundamental mathematical disciplines, such as Complex Analysis, Differential Geometry, and Partial Differential Equations (PDEs). From the point of view of Complex Analysis, CR- geometry is a tool for studying holomorphic functions in several variables; from the point of view of Differential Geometry, CR-geometry is a model geometry in the framework of Cartans Moving Frame machinery; from the point of view of the theory of Linear PDEs, CR-geometry is a tool for studying general linear PDEs by means of geometric methods coming from Complex Analysis and Geometry. In his recent work, the PI have discovered a new face of CR-geometry, as being closely connected to the theory of (continuous) Dynamical Systems. The PI, in his work with several co-authors, have developed a vocabulary between objects of study of CR- geometry (CR-manifolds and CR-maps) on one hand, and classes of dynamical systems and their transformations on the other hand. This bridge technique is called the CR DS technique. The CR DS technique has recently enabled to solve a number of long- standing problems concerning CR-manifolds with strong degeneracies of the CR-structure on them. The current research project is aimed to address by means of the CR DS technique further difficult problems in CR-geometry, this time not only related to mappings of degenerate CR-manifolds, but also to those between nondegenerate CR-manifolds of different dimensions. It is planned to investigate the Gevrey regularity of mappings between degenerate real hypersurfaces in complex space, investigate the analytic regularity of CR-embeddings between strictly pseudoconvex real hypersurfaces, obtain necessary and sufficient conditions for the embeddability of CR-manifolds into a hyperquadric, and algebraizability conditions for real hypersurface. The project team will include the PI, Dr. Jan Gregorovic, and collaborating colleagues from the Vienna Institute of Technology.