Ilya Kossovskiy
Title: Dynamical Approach to CR-geometry and Applications

Abstract: Study of equivalences and symmetries of real submanifolds in complex space goes back to
the classical work of Poincar\'e  and Cartan  and was deeply developed in later work of Tanaka
and Chern and Moser. This work initiated a large number of further publications in the area
(since 1970's till present), which are dedicated to the questions of regularity of mappings
between real submanifolds in complex space, unique jet determination of mappings, solution of the
equivalence problem, and study of automorphism groups of real submanifolds. Current state of the
art and methods involved provide satisfactory (and sometimes complete) solution for the above
mentioned problems in nondegenerate settings.

However, very little is known for more degenerate situations, i.e., when real submanifolds under
consideration admit certain singularities of the CR-structure (such as non-constancy of the
complex tangent dimension or that of the CR-orbit dimension).


The recent CR (Cauchy-Riemann Manifolds) -- DS (Dynamical Systems) technique, developed in our
joint work with Shafikov and Lamel, suggests to replace, in a certain way, CR-singular real
submanifolds by appropriate complex dynamical systems. This technique has recently demonstrated
the ability to solve a number of long-standing problems, related to mappings of CR-singular real
submanifolds. 


In this talk, we give an overview of the technique and the results obtained recently by using it.

We also discuss a possible development in this direction, in particular, new extension phenomena
for CR-mappings.