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 Okatheory, AndersénLemperttheory 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 Okatheory is concerned with flexibility properties of holomorphic maps from Stein manifolds (natural sources for holomorphic maps) to Okamanifolds ("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 Okatheory, 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 AndersénLemperttheory is a relatively recent tool/topic that emerged during the early 1990's following a fundamental paper of RosayRudin 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 coauthors. 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 preplanned 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 noncurrent Fatoucomponents, 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.
Fellows

Arosio, Leandro

Bracci, Filippo

Fornæss, John Erik

Forstneric, Franc

Globevnik, Josip

Kutzschebauch, Frank

Larusson, Finnur

Lempert, Laszlo

Løw, Erik

Peters, Han

Shcherbina, Nikolay

Vivas, Liz
Previous events

19 Jun  23 Jun 2017Hotel Gabelshus, Oslo Hotel Gabelshus, Oslo

30 May 201713:15  14:30Turret Room, CAS Oslo Turret Room, CAS Oslo

27 Apr 201713:15  16:00Turret Room, CAS Oslo Turret Room, CAS Oslo

30 Mar 201713:15  16:00Turret Room, CAS Oslo Turret Room, CAS Oslo

09 Mar  12 Mar 2017(all day)To be announced To be announced

16 Feb 201713:15  16:00Turret Room, CAS Oslo Turret Room, CAS Oslo

09 Feb 201713:15  16:00Turret Room, CAS Oslo Turret Room, CAS Oslo

26 Jan 2017

08 Dec 201613:15  16:00Turret Room, CAS, Oslo Turret Room, CAS, Oslo

24 Nov 201613:15  16:00Turret Room, CAS, Oslo Turret Room, CAS, Oslo

17 Nov 201613:15  16:00Turret Room, CAS, Oslo Turret Room, CAS, Oslo

03 Nov 201613:15  15:00Turret Room, CAS, Oslo Turret Room, CAS, Oslo

31 Oct 201612:00  13:00The Turret Room, Centre for Advanced Study The Turret Room, Centre for Advanced Study

10 Oct  14 Oct 2016(all day)Torbjørnrud Hotel, Jevnaker Torbjørnrud Hotel, Jevnaker

29 Sep 201613:15  15:00CAS, Oslo CAS, Oslo

22 Aug  26 Aug 2016(all day)DNVA DNVA
News

Alumni Spotlight: Berit Stensønes about her role model Karen Uhlenbeck
27.03.2019 
– Complex numbers make the world bigger
26.04.2017 
‘Spaces of Spaces’
02.11.2016
Group leader
Publications
 Alarcón, A. & F. Forstneric. 2017. "Darboux charts around holomorphic Legendrian curves and applications."
 Arosio, L. et al. 2017. "Dynamics of Trancendental Hénon Maps."
 Arosio, L. et al. 2017. "Squeezing functions and Cantor Sets."
 Arosio, L. & E. F. Wold. 2017. "Totally real embeddings with prescribed polynomial hulls."
 Bracci, F. & H. Gaussier. 2017. "The proof of the MuirSuffridge conjecture."
 Fornæss, J. E. & E. F. Wold. 2016. "A nonstrictly pseudoconvex domain for which the squeezing function function tends to one towards the boundary."
 Forstneric, F. 2016. "Surjective holomorphic maps onto Oka manifolds."
 Forstneric, F. & F. Larússon. 2017. "The Oka principle for holomorphic Legendrian curves in C2n+1. "
 Kutzchebauch, F., F. Larússon & G. Schwarz. 2016. "An equivariant parametric Oka principle for bundles of homogeneous spaces."
 Kaliman, S., F. Kutzschebauch & T. T. Truong. 2017. "On subelliptic manifolds."
 Lempert, L. 2017. "Isometries in spaces of Kähler potentials."
 Lempert, L. 2017. "On Complex Legendre duality."
 Lempert, L. 2017. ”Riemannian geometry in infinite dimensional spaces.”
 Sibony, N. & E. F. Wold. 2016. "Topology and complex structures of leaves of foliations by Riemann surfaces."
 Simon, L. & B. Stensønes. 2017. "On Newton diagrams of plurisubharmonic polynomials."
 Wold, E. F. 2016. "Asymptotics of invariant metrics in the normal direction and a new characterisation of the unit disk."