Centre for Advanced Study

at the Norwegian Academy of Science and Letters

Several Complex Variables and Complex Dynamics


Former 2016/2017 Natural Sciences - Medicine - Mathematics


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.

End Report

Mathematicians seldom work in large groups, and typically they work in groups of two or three. The CAS concept is a great one; it gives groups a chance to focus entirely on research for more than a week or two, without being interrupted by teaching and administrative duties. A key point is the extended time: one thing is to meet and to carry out pre-planned projects, this can often be carried out with the aid of shorter meetings and technology. Being at the same center over an extended period of time, one the other hand, allows for synergy effects that are impossible to plan in advance. Questions and ideas are allowed to float freely amongst the participants, and this leads to a new level of creativity.

The year was very productive, and major progress was made in a variety of directions we had already planned, but also in new and unexpected ones. Some of the topics the project made progress in relates to non-current Fatou-components, invariant metrics, the ∂-equation, injective holomorphic mappings, transcendental Hénon maps, and infinite dimensional complex analysis. Please read about all of the projects findings here (PDF).

In addition, several projects were started during the year, but has not yet reached the maturity to be presented in any detail. Towards the end of the year, several CAS fellows were inspired to look for further possibilities to continue this fruitful interaction, including an application for an ERC synergy grant.



  • Arosio, Leandro
    Post doc University of Rome 2016/2017
  • Bracci, Filippo
    Professor University of Rome, Tor Vergata 2016/2017
  • Fornæss, John Erik
    Professor Norwegian University of Science and Technology (NTNU) 2016/2017
  • Forstneric, Franc
    Professor University of Ljubljana 2016/2017
  • Globevnik, Josip
    Professor Em. University of Ljubljana 2016/2017
  • Kutzschebauch, Frank
    Professor University of Bern 2016/2017
  • Larusson, Finnur
    Professor University of Adelaide 2016/2017
  • Lempert, Laszlo
    Professor Purdue University 2016/2017
  • Løw, Erik
    Professor University of Oslo 2016/2017
  • Peters, Han
    Associate Professor University of Amsterdam 2016/2017
  • Shcherbina, Nikolay
    Professor University of Wuppertal 2016/2017
  • Vivas, Liz
    Assistant Professor Ohio State University 2016/2017

Previous events


Group leader

  • Erlend Fornæss Wold

    Title Professor Institution University of Oslo (UiO) Year at CAS 2016/2017
  • Berit Stensønes

    Title Professor Institution Norwegian University of Science and Technology (NTNU) Year at CAS 2016/2017