Several Complex Variables and Complex Dynamics

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Former 2016/2017 Natural Sciences - Medicine - Mathematics

Abstract

This project is a cooperation between the Several Complex Variables (SCV) groups at UiO and at NTNU, as well as an extensive international network consisting of the international expertise on the proposed topic: Interactions between Oka-theory, Andersén-Lempert-theory and Complex dynamics, with applications in complex geometry, and the geometry of real analytic boundaries and the ∂-equation.

Over the recent years we have seen an extremely fruitful interplay between the above mentioned topics, with a wide number of applications in SCV, and other areas of mathematics/science, such as low dimensional topology, algebraic geometry and accelerator physics. At the same time, classical topics in SCV, such as quantitative solutions to the -equation, are fundamental for the development of these new machineries. There is reason to believe that we have only seen the very beginning of the development and applications of these powerful tools, and we feel that the time is right to gather some key actors on the world scene, so that we can work together at the same place over a substantial period of time.

The core participants, in addition to the project leaders, are John Erik Fornæss (NTNU, Trondheim), Franc Forstneric (University of Ljubljana), Frank Kutzschebauch (University of Bern), Filippo Bracci (University of Rome, Tor Vergata) and Erik Løw (UiO).

 

The Oka-theory is concerned with flexibility properties of holomorphic maps from Stein manifolds (natural sources for holomorphic maps) to Oka-manifolds ("good" targets for holomorphic maps from Stein manifolds), and provides many powerful tools for constructions in complex geometry and complex dynamics.  Our collaborators Franc Forstneric and Finnur Larusson are of the main developers of the modern Oka-theory, revived by M. Gromov in the late 1980's.  The theory has its roots in work of K. Oka and the H. Gauert school, and constitutes one of the main pillars of SCV.  

The Andersén-Lempert-theory is a relatively recent tool/topic that emerged during the early 1990's following a fundamental paper of Rosay-Rudin in 1988. Work of our collaborator Forstneric and J. P. Rosay in 1994 exhibited the extreme flexibility of the automorphism group of complex euclidean space, thereby furnishing a powerful tool in complex geometry, and has been at the core of a well of results over the last twenty years.  Our collaborator Kutzschebauch is one of the main developers and utilizers of the theory, and with collaborators he has lifted the theory also to the realm of affine algebraic geometry.    

In Complex Dynamics one studies iterations of holomorphic mappings (discrete dynamics) or evolutions of holomorphic vector fields (continuous dynamics).  Whereas  many topics in general dynamical systems are currently far out of reach, the structure imposed by holomorphicity allows for strong results, and makes the topic phenomenologically very important for the study of dynamics in general.  One of the key developers of dynamics in several complex varibles, starting in the mid 1980's, is our collaborator John Erik Fornæss, who is also one of the world leading experts on SCV.     

Loewner Theory in one variable is an old subject which proved to be a cornerstone in geometric function theory.  For instance it is one of the mail tools in deBrange's proof of the Bieberbach conjecture, and Shramm's SLE's theory.  In higher dimensions a general theory has been developed on complete hyperbolic manifolds by our collaborator Filippo Bracci and his co-authors.  Applications to geometry of domains, dynamics and univalent mappings in higher dimensions are currently under investigation.

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