Symposium on Complex Analysis and Complex Dynamics

Discrete Complex Dynamics: The theory has developed fast in the last 20 years with the advent of computers. The theory consists of the Fatou theory of orderly behaviour and the Julia theory of chaotic behaviour. There has been many advances but many basic problems remain unsolved. Of these we can mention whether two competing notions of Julia sets for Henon maps are equivalent. In the Fatou theory, there are questions about classification of Fatou sets.

Continuous Dynamics: This arises in the study of polynomial vector fields. The integral curves are Riemann surfaces which foliate space. The foliation has singular points. There has been much work on understanding the global behaviour of the leaves, such as showing that under some hypotheses, leaves are dense. The use of positive harmonic currents introduced by Fornæss and Sibony has been very fruitful. A main question left is that of existence of minimal exceptional sets, arising from taking the closure of a leaf which does not cluster at any singular point. Hyperbolicity: This classical notion introduced by Kobayashi is a very active area of research. Important advances have been made in the last period, among them Siu’ work on hyperbolicity of a generic algebraic hypersurface of large degree in Cn or CPn . The methods of Demailly also play major role. The Green-Griffiths conjecture concerning almost-hyperbolicity of compact complex manifolds of general type is still a major challenge.

Holomorphic flexibility and modern Oka theory: Opposite to hyperbolicity, flexibility properties of a complex manifold signify that it admits many holomorphic maps from Euclidean spaces and other Stein manifolds. The central notion is that of an Oka manifold (Forstneriˇc 2009) which evolved from the classical Oka-Grauert principle. Among the main sufficient condition are Gromov’s ellipticity and flexibility of Arzhantsev et al. This field offers challenging problems and applications. Recent progress includes a study of directed immersion of Riemann surfaces, with emphasis on null curves and their applications to the classical Calabi-Yau problem on minimal surface.

Invariant theory in Complex Analysis: This arises when one tries to understand when are two objects holomorphically equivalent. A basic approach is to attach holomorphic invariants to the objects such as hypersurfaces. The geometric invariants are constructed using the Cartan theory, while the algebraic invariants are accessible through the Poincare-Moser normal form theory. The subject has been intensively studied in the past many years and has applications to many other problems in complex analysis.

Complex Plateau Problem and singularity theory: Singularity theory plays a very important role in algebraic geometry, differential geometry, topology, dynamic system, etc. This field offers many challenging problems among which the complex Plateau problem is one of the important ones. This problem forms an intrinsic link between CR geometry and singularity theory. The invariants theory of singularities, developed by S. Yau, has been proven very fruitful. A main open question is the interior regularity for non-hypersurface type strongly pseudoconvex CR manifolds; this is also related to the famous Hartshorne conjecture in algebraic geometry.

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Shiu Yuen ChengTsinghua University, China
John Eric FornæssNorwegian University of Science and Technology
Franc ForstnericUniversity of Ljubljana, Slovenia
Xiaojun HuangRutgers University, USA
Stephen S.T. YauTsinghua University, China
Xiangyu ZhouChinese Academy of Sciences, Beijing