Projects / Programmes
January 1, 2022
- December 31, 2027
| Code |
Science |
Field |
Subfield |
| 1.01.00 |
Natural sciences and mathematics |
Mathematics |
|
| Code |
Science |
Field |
| 1.01 |
Natural Sciences |
Mathematics |
holomorphic map, Stein manifold, Oka manifold, minimal surface, holomorphic Legendrian curve, complex contact manifold, Einstein manifold, CR singularity, holomorphic dynamics, Fatou component, Fourier transform, p-ellipticity, partial differential equation, metric graph
Data for the last 5 years (citations for the last 10 years) on
April 24, 2024;
A3 for period
2018-2022
| Database |
Linked records |
Citations |
Pure citations |
Average pure citations |
| WoS |
412 |
3,998 |
2,908 |
7.06 |
| Scopus |
400 |
4,051 |
2,913 |
7.28 |
Researchers (20)
Organisations (3)
Abstract
We shall investigate several interrelated groups of problems in complex analysis, complex geometry, complex dynamics, Fourier analysis, partial differential equations, and analysis on infinite metric graphs. In the field of complex analysis and geometry we shall study questions related to the class of Oka manifolds, their position in complex geometry with respect to other standard classes of manifolds, and we shall continue developing Oka-theoretic methods for applications in the theory of minimal surfaces, in complex contact geometry, and elsewhere. We shall look into the possibility of providing a geometric characterization of the class of Oka domains with compact complements in Cn, the union problem for Oka manifolds, Oka properties of elliptic surfaces, the connection between Oka manifolds and Campana special manifolds, their relationship with metric positivity of compact projective and Kaehler manifolds, and the study of degenerations of Euclidean fibres in holomorphic Stein fibrations. We shall investigate the Calabi-Yau problem for minimal surfaces in nonstandard geometries. We will find bounds on derivatives of conformal minimal surfaces in model domains and develop the hyperbolicity theory of domains in Rn by way of minimal surfaces. We shall develop the theory of minimal hulls of compact sets in Rn. We will look for new geometric invariants of holomorphic contact structures on complex Euclidean spaces. We will try to solve the existence problem for algebraic Kobayashi hyperbolic contact structures on C3. We will adapt the methods of Oka theory to obtain new construction techniques for holomorphic Legendrian curves. We shall investigate the possibilities offered by the Penrose theory of twistor spaces in the interplay between superminimal surfaces in self-dual Einstein 4-manifolds and complex contact 3-manifolds. We will study Cauchy-Riemann singularities of real submanifolds in complex manifolds and their normal forms in codimension 2. We will study multidimensional complex dynamics with emphasis on the analysis and classification of wandering Fatou components of holomorphic automorphisms of Euclidean spaces, endomorphisms of complex projective spaces, and local dynamics of holomorphic maps tangent to the identity. We shall develop new discretization methods for computing the inverse nonlinear Fourier transform by way of superposition of elementary model solutions. We shall investigate the role played by our new p-ellipticity condition which we have introduced in the study partial differential equations. In particular, we shall focus on the role of p-ellipticity in the development of dimensionless estimates for bilinear and trilinear embeddings. We shall develop new analytic methods in the study of infinite metric graphs - the quantum graphs - with emphasis on non-locally finite graphs.
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 mathematical journals, including the most distinguished ones. We can be proud of our accomplishments in the period 2015-2020 when we succeeded in placing a series of our works in the most elite mathematical journals such as Inventiones Math., Duke Math. J., JEMS, Memoirs Amer. Math. Soc., American J. Math., Analysis & PDE, Geom. & Topol., Proc. London Math. Soc., Math. Ann., and several others. We also published two monographs with Springer-Verlag. It is expected that many of our results will find future applications both within mathematics (where applications have already been found) as well as in other fields of science, and they will form a background for improvements in certain applied areas such as technical sciences, computer science, informatics, pharmacology, 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. By keeping the highest standards of mathematical development, our research group projects 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 at some of the most 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, Geometry and partial differential equations, which are all topics that we study. We can make this statement more concrete - the programme coordinator, Franc Forstnerič, received the Stefan Bergman Prize for 2019. This major international prize, administered by the American Mathematical Society, is given to researchers in the fields of complex analysis, harmonic analysis, and operator theory. This was only the 8th such prize for a mathematician from Europe since its inception in 1989. Another major recognition was the invitation to Forstnerič as one of 10 plenary speakers at 8th European Congress of Mathematics in 2021. In the last 5 years, Oliver Dragičević made fundamental and lasting contributions to the theory of elliptic partial differential equations. Together with collaborator Andrea Carbonaro from University of Genoa, he introduced in the literature the notion of p-ellipticity, generalizing the standard notion of ellipticity. They have amply demonstrated the relevance of this notion by using it to give a complete and precise solution of a long-standing open problem in a couple of papers in prestigious journals (Duke Math. J. 2017, JEMS 2020; precise data are given in the section on our main achievements). This notion has already attracted attention of important researchers worldwide and is now being used in diverse problems. Based on these developments, Dragičević received invitation to give a plenary lecture at the conference "Analysis and Applications" to be held in Wrocław (Poland) in 2021. Among five members of the programme committee of the said conference, there are two Fields medalist, Charles Fefferman and Terence Tao, two of todays's most prominent analysts. The standing of Dragičević in the international harmonic analysis community can also be illustrated by the following fact. Within the scope of the 8th Europ. Congress of Mat. 8ECM in 2021, he is organizing a satellite conference as well as a minisymposium. His invitation to speak at these events was accepted by several of today's most prominent analysts worldwide. The main speaker on his minisymposium will by Jill Pipher, currently serving as president of the Amer. Math. Soc. Another group member, Aleksey Kostenko, is deemed to be one the most important analysists of his generation in the fields which he pursues, which include spetral theory of operators and more recently analysis on infinite graphs. He is currently in the process of promotion to a full professor at University of Ljubljana and has received enthusiastic letters of support from some of the leading foreign experts in these fields.
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 its intellectual and socio-economic goal. The basis for this and for all advanced technological areas of economic 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. In the past years, a major effort went to education of young researchers. Four of our PhDs which completed the program in the last years came from abroad (Italy, Australia, Austria); one remained in Slovenia employed in private sector, andn the other ones are postdoctoral researchers at different universities. Converesely, several students who obtained undergraduate or Master diplomas with members of our group as advisers are now working on PhD at major foreign universities such as Univ. of Zurich, Univ. of Canberra, Univ. of Texas-Austin, and others. Our research activities also had a direct impact on our teaching. Most notably, Forstnerič has in collaboration just published a monograph with Springer on the subject of minimal surfaces, and it is expected that parts of it will be useful in preparation of new courses in this field.
