Projects / Programmes
January 1, 2004
- December 31, 2008
| Code |
Science |
Field |
Subfield |
| 1.01.00 |
Natural sciences and mathematics |
Mathematics |
|
| Code |
Science |
Field |
| P130 |
Natural sciences and mathematics |
Functions, differential equations |
Stein manifolds, holomorphic maps, complex manifolds, proper maps, Stein domains, complex points, strictly pseudoconvexconvex hypersurfaces, plurisubharmonic functions, argument principle, h-principle, analytic extensions, integrable systems, isometries of Banach spaces, Riesz transforms.
Researchers (17)
Organisations (1)
Abstract
We intend to perform research and obtain new results in thirteen areas in complex analysis and geometry, global analysis, functional analysis and geometry, and harmonic analysis. These areas are:
1. Complex analysis and geometry
- Study of nonlinear boundary value problems for quazilinear equations on finite Riemann surfaces
- Construction of proper holomorphic maps from the disc whose ranges miss a given pluripolar set
- Construction of regular holomorphic maps from Stein manifolds to complex manifolds
- Construction of differentiable Cauchy-Riemann embeddings of strictly pseudoconvex hypersurfaces into spheres
- Construction of Stein domains in non-Stein complex manifolds
- Analysis of complex analytic properties of real surfaces embedded into complex surfaces
- Morse theory of plurisubharmonic functions
- Boundaries of analytic sets and the argument principle
- H-principle for modified Stein Spaces
- Analytic extensions from families of curves
2. Global analysis
- Singularity structure of integrable systems
3. Functional analysis and geometry
- Isometries of Banach spaces and special norms on finite-dimensional spaces
4. Harmonic analysis
- Riesz transforms
Significance for science
During the years 2004-2008 the work of the group of the program »Complex Analysis and geometry« produced important new results in various areas in complex analysis and geometry, global analysis and geometry and harmonic analysis which were published in international mathematical journals. In complex analysis and geometry we obtained new results about holomorphic curves in complex spaces, the Oka-Grauert-Gromov principle, holomorphic embeddings, immersions and submersions, about the existence of open Stein neighbourhoods, about normal forms of almost complex structures, about nonlinear Riemann-Hilbert problems, about boundary values of holomorphic functions related to the argument principle. In global analysis we obtained results about Maxwell-Bloch equations and in harmonic analysis we obtained results are about L-p estimates for Hilbert and Ahlfors-Beurling operators. During the years 2004-2008 the members of the group published 52 papers in mathematical journals covered by SCI, 12 of which were written in collaboration with other mathematicians. Some of these papers were long and published in the leading Journals such as Annals of Mathematics, Duke Journal of Mathematics. As the most important results we want to mention - a new technique of constructing holomorphic mappings with a technique of gluing holomorphic sprays (Drinovec-Drnovšek, Forstnerič) which resulted in optimal results about proper holomorphic images of a bordered Riemann surface - the proof that the classical Oka property in a complex manifold is equivalent to the convex approximation property (Forstnerič) - results about solutions of Riemann-Hilbert problems on bordered Riemann surfaces (Černe) - constructing a new Hamiltonian structure of the Maxwell-Bloch equations (Saksida) - a rotation method which gives L-p estimates for powers of the Ahlfors-Beurlig operator (Dragičević). Several members of the group have been collaborating with mathematicians from other countries. Several members of the group gave invited lectures and talks at international conferences and gave lectures at various universities throughout the world In 2006 the members of the group organized a successful international conference in Kranjska Gora »Symposium in complex analysis – Slovenia 2006« . The list of invited speakers included some of the most prominent names in the world in this area and it attracted an excellent group of researchers in in the field.
Significance for the country
Mathematics is a basis in many fields of research. High quality mathematical research is a basis of educating mathematicians needed at all four universities in Slovenia and other institutions of higher education. The research results of the program »Analysis and geometry« in the years 2004-2008 will be potentially applicable for researchers in these areas everywhere in the world and consequentially for all people in Slovenia interested in these areas. Several members of the research team are regularly invited as speakers at international meetings and at various universities. Within this program we organized in 2006 an international meeting on complex analysis and geometry in Slovenia. The list of invited speakers included some top names in the world from this area. The research team is of high quality and well known in the world. Our regular frequent working contacts with best researchers in the world will continue into the future. Thus, in mathematical research we contribute to streghtening the national identity of Slovenia and the international recognition of Slovenia.
Most important scientific results
Final report,
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Most important socioeconomically and culturally relevant results
Final report,
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