Dynamics and CR Geometry

We propose to study problems from local dynamics, CR geometry, algebraic geometry, and pluripotential theory which are closely linked to each other: Local dynamics helps to reduce classification of symmetric CR manifolds; these, on the other hand, appear as natural invariant objects for some dynamical systems. The question surrounding the difference between an analytic and a formal classification leads to an exciting link with open problems surrounding the Artin approximation theorem in algebraic geometry.Lastly, in rougher settings, one needs to find substitutes for analytic geometric invariants, and we propose to look for them in pluripotential theory.

Our project aimed at bridging two distinct areas of mathematics, namely Cauchy-Riemann (CR) geometry on the one hand and the theory of Dynamical Systems on the other hand. We cooperated with a group from the University of Nice Sophia Antipolis, headed by Prof. Laurent Stolovitch, whose main expertise lies in the theory of dynamical systems. The group from Vienna specializes in CR geometry. The main interest which drives us to explore this connection is the interplay between "formal" and "real" mappings. Whenever one tries to decide whether two given objects (dynamical systems or CR manifolds) are actually the same, or whenever one tries to find simple representations of a given object, one encounters very easily and naturally the formal mappings occuring in this problem. However, one is usually interested in providing not only formal, but real mappings solving the problem. Our objects, dynamical systems, are natural mathematical models for many natural phenomena, while CR manifolds encode in a geometric way aspects of certain partial differential equations. Our project was very successful: Its outcomes have been published in some of the leading mathematical journals world-wide (such as Acta Mathematica and Inventiones Mathematicae). Two types of results should be mentioned in this quick summary: First, we studied "convergence" properties of formal maps. Convergence ensures that a formal map is really a map (in a very strong way). One of our main results in that area shows that the known natural obstruction to convergence is actually the only obstruction: If one has a divergent holomorphic map into a CR manifold, that CR manifold necessarily contains complex subvarieties (you can roughly think about them as regions where the structure breaks down). Furthermore, we successfully described simple models (so-called normal forms) for a large class of CR manifolds, which allow one to compare these CR manifolds. Even though we gave formal normal forms, we could establish an especially simple and elegant criterion ensuring the convergence of these normal forms (of Levi-nondegenerate real submanifolds).