Projects / Programmes
Holomorphic mappings, geometric interpolation, and harmonic analysis
| 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 mappings, the Oka-Grauert principle, harmonic analysis, Brauer groups, geometric interpolation
Researchers (12)
Organisations (2)
Abstract
We shall investigate several important open problems on the intersection of complex, harmonic and numerical analysis and algebra, with complex analysis as the main unifying theme connecting the various parts of the proposal.
We shall continue our research on Oka-Grauert-Gromov principle and its applications in analytic geometry. We will study the allowable operations on the class of Oka manifolds and the question which complex surfaces and higher dimensional manifolds belong to this class. We shall analyze the plurisubharmonicity properties of envelopes of several classical disc functionals on complex spaces with singularities. Finally, we shall study the geometry of Cauchy-Riemann (CR) submanifolds in complex manifolds, with emphasis on CR singularities (complex points) in real codimension two submanifolds. In particular, we will try to find and classify normal forms of elliptic points in real codiimension two submanifolds, thereby extending the classical work done for surfaces.
Complex analysis has played a prominent role in the invariant theory of finite and algebraic groups, a classical part of algebra developed by D.Hilbert and E. Noether in the beginning of 20th century. We will study the unramified Brauer groups and higher unramified cohomology of finite and algebraic groups. Using the homological approach we have developed, we will revisit the theory of minimal factors developed by Bogomolov. One of our main goals will be a complete classification of minimal factors. Another direction will be the study of unramified Brauer groups of finite p-groups of given coclass. Our conjecture is that, given a prime p and integer r, there exist only finitely many finite p-groups of coclass r with trivial unramified Brauer group. In addition, we will determine unramified Brauer groups of powerful p-groups which are key constituents of general p-groups. Consequently, we will obtain refined obstructions to Noether's problem. Beside that, we will initiate the study of minimal factors with respect to higher unramified cohomology.
In harmonic analysis we will study exact estimates of powers of the Ahlafors-Beurling operator T which is important in qusiconformal mapping theory. We will also attempt proving the bilinear embedding theorem for the Kohn-Laplacian on Heisenberg-type groups, consequently deriving concrete Lp estimates for the Heisenberg-Riesz transforms. It is known that these estimates are dimension-free. It is also true that one can merge them with results pertaining to the parabolic Hilbert transform and thus prove an explicit growth of norms for p → ∞, namely of order p1+ε for arbitrary ε)0. By applying the method of Bellman functions we would attempt to prove linear growth, which would represent a circumstantial evidence in favour of an old conjecture regarding weak-type 1-1 bounedness of the parabolic Hilbert transform.
In numerical analysis we will carry on the research of the geometric interpolation by polynomial and rational parametric curves. Of a particular interest will be interpolation problems that are important when designing solid body movement. Efficient solutions of such problems are an important issue in the fields such as the robotics etc. Among the questions considered will be geometric Lagrange interpolation shemes based upon low degree rational curves that admitt a closed form solutions. Such a sheme would lead to a practically impotant nonlinear interpolatory subdivision algorithm.A part of the research will be devoted to the construction of parametric curves with a rational frame, in particular the rotation minimizing one. Here an important role is played by P-curves, and the more familiar Pythagorean-hodograph (PH) curves.
Significance for science
Work on this project led to important original scientific discoveries in the field of complex analysis and geometry, theory of minimal surfaces, harmonic analysis, integrable Hamiltonian systems, algebra and numerical analysis. The results were properly explained in detail and documented through publications in international scientific journals with impact factor. Main results were also disseminated through numerous invited lectures at international conferences, including some plenary lectures, through courses at several international PhD schools, and by lectures at foreign Universities and research institutions. The results achieved in the scope of this project gained international recognition as can be seen by citations, and also by numerous and regular invitations to lectures at international conferences and foreign research institutions. All this contributed to a wider visibility and accessibility of our research achievements, in particular among younger generations of researchers worldwide. We have also actively engaged in popularization of science.
Significance for the country
Our scientific achievements compare very favorably with those of leading research groups in our field, both in Europe and worldwide. For this reason our research work substantially contributed to the international visibility and recognition of Slovenia in the field of mathematical sciences. The new knowledge that was achieved through work on this project was continuously used and transferred to our working environment, mainly through lectures at all levels of studies at the University of Ljubljana, but also through organization of scientific seminars, visits of foreign experts and lecturers, organization of research conferences both at home and abroad, etc. Through these and other activities we are continuously striving to create stimulating environment for the ripening of new ideas and for the education and development of new generations of engineers, teachers, professor and researchers. When necessary and appropriate, we also actively participate in social aspects of the development and maintenance of education and science, in particular by expressing opinions in occasionally also by taking leading roles and positions in our institutions. Through our editorial work at domestic and international journals we contributed to the development and progress in education and science, both in Slovenia and worldwide.
Most important scientific results
Annual report
2013,
2014,
2015,
final report
Most important socioeconomically and culturally relevant results
Annual report
2013,
2014,
2015,
final report
