Projects / Programmes
January 1, 2015
- December 31, 2021
| Code |
Science |
Field |
Subfield |
| 1.01.00 |
Natural sciences and mathematics |
Mathematics |
|
| Code |
Science |
Field |
| P001 |
Natural sciences and mathematics |
Mathematics |
| Code |
Science |
Field |
| 1.01 |
Natural Sciences |
Mathematics |
holomorphic functions, complex manifolds, minimal surfaces, Fourier series, Riesz operators, integrable systems, Lie grupoids
holomorphic functions, complex manifolds, Stein manifolds, Oka principle, minimal surfaces, Fourier series, Riesz operators, Hamiltonian systems, Lie grupoids
Researchers (23)
Organisations (2)
Abstract
We are proposing research on contemporary topics in complex analysis and geometry, Oka-Grauert-Gromov theory, Cauchy-Riemann geometry, pluripotential theory, minimal surfaces, harmonic and Fourier analysis, partial differential equations and relations, theory of integrable systems, mathematical physics, invariants of Lie grupoids and algebroids.
The proposal is to a certain extent a continuation of our past and current research on topics where we gave several major contributions as is evident from our publications in leading mathematical journals during 2009-14. In addition, the proposal contains several new directions of investigation. We mention in particular new applications of the Oka-Grauert-Gromov theory, a subject that was brought in a coherent form during the last decade and is summarized in the monograph F.Forstnerič, Stein Manifolds and Holomorphic Mappings, Springer-Verlag (2011). It recently became clear that our methods are very useful in the classical theory of minimal surfaces, holomorphic null curves, and other objects represented by directed immersions of Riemann surfaces. This opens a new dimension in the theory of first order holomorphic partial differential relations. We expect major new applications of our method of gluing holomorphic sprays and of exposing boundary points. We shall investigate the connections between the theory of minimal surfaces and of null-plurisubharmonic functions in the general framework of Harvey-Lawson theory. Globevnik will continue his investigations concerning the extendibility of holomorphic functions, a topic on which he is a leading expert. We expect to obtain new results on boundary differential relations for holomorphic function on finitely connected planar domains. With the addition of U.Kuzman (PhD 2013) our research is expanding into almost complex geometry. O.Dragičević will focus on sharp estimates for spectral multipliers and sharp dimension-free bilinear estimates for operators in divergence form with complex coefficients. Slapar will work on the classification of normal forms of complex points in real submanifolds, an essential ingredient for understanding local polynomial hulls, and on the topological characterization of q-convex manifolds. P. Saksida will apply his results on nonlinear Fourier transform to the analysis of the sine-Gordon equation and to the nonlinear Schrödinger equation. Mrčun, Jelenc and Kališnik will study the homotopy theory of foliation groupoids and homology groups of certain Lie groupoids, geometric description of homology and cohomology of topological groupoids, and connections on Lie groupoids which are used to study their representations. We are planning to collaborate with several foreign partners, notably A. Alarcon and F. Lopez (University of Granada), F. Larusson (University of Adelaide), E.F.Wold and T.Ritter (University of Oslo), F.Kutzschebauch (University of Bern), A. Sukhov (University of Lille), A. Volberg (University of Michigan, East Lansing), A. Carbonaro (University of Genova).
Significance for science
It is expected that the results obtained under the proposed research program will represent substantial and highly nontrivial contributions to the mathematics in all proposed areas of research, with applications to mathematical physics and other fields of science. They will be published in international scientific journals. Continuing our long standing tradition, we shall strive to publish high level work in leading journals in the respective fields, including the most distinguished ones. We can be proud of our accomplishments in the period 2009-14 when we succeeded in placing a series of our works in the most elite mathematical journals such as Inventiones Math., Duke Math. J., Advances in Math., American J. Math., Analysis & PDE, J. Anal. Pures Appl., J. Funct. Analysis. It is expected that many of these results will find future applications in other fields of science, and they will form a background for improvements in certain applied areas such as technical sciences, computer science, informatics, farmakology, mechanical engineering, and others. We shall continue applying our research achievements to improve and enrich our pedagogical work, especially in the areas of masters and doctoral studies at University of Ljubljana. The proposed research work will also provide ample opportunity to educate a new generation of young researchers.
Significance for the country
High quality education and top level scientific development are the largest Slovenian priority and challenge. As a small country without major natural resources, it must depend on its development capacities and human potential in reaching itse intelectual and socieconomic goal. The basis for this and for all advanced technological areas of economical development is a high level scientific work. Mathematics is universally useful and omnipresent in today's world; major mathematical discoveries sooner or later find their way into everyday life. Mathematics provides major direct contributions to the development and improvements in areas of applied natural and technical sciences (computer science, informatics, engineering, farmacology, etc.), and is the best available tool for development of human mind, logical thinking and problem solving. Teachers who are educating new generations of students and researchers can be only as good as their scientific work. By keeping the highest standards of mathematical developments we project to the world the image of the high quality of our mathematics. Our scientific environment is becoming increasingly more attractive in international circles as is shown by numerous invitations received by members of our team for visits at even the best world universities, invitations to lectures prestigious international conferences, and frequent visits of distinguished foreign researchers to our group. The relevance and importance of mathematical fields, pursued by our group, is shown by the fact that in recent years, most of the highest international prizes in Mathematics went to researchers in Analysis and Geometry.
Most important scientific results
Annual report
2015,
interim report
Most important socioeconomically and culturally relevant results
Annual report
2015,
interim report
