Projects / Programmes
January 1, 2004
- December 31, 2008
| Code |
Science |
Field |
Subfield |
| 1.01.00 |
Natural sciences and mathematics |
Mathematics |
|
| Code |
Science |
Field |
| P150 |
Natural sciences and mathematics |
Geometry, algebraic topology |
geometric topology, algebraic topology, manifold topology, piecewise-linear topology, infinite-dimensional topology, fractal geometry, differential geometry, wild Cantor sets, Peano continua, cell-like mappings, 3-manifolds, 4-manifolds, knot theory, cohomological dimension, Lie groups, homotopy selfequivalences, CW complexes, symplectic topology.
Researchers (19)
Organisations (1)
Abstract
The proposed five year research program will focus on some very difficult problems from modern topology and geometry. In particular, in geometric topology we shall consider topology of Peano continua in the plane, cell-like mappings on 4-manifolds and problems related to cohomological dimension of compact metric spaces. We shall also study the questions of homogeneity and rigidity of wild Cantor sets in Euclidean spaces of dimension greater than two. In algebraic topology we shall investigate homotopy autoequivalences and CW homotopy types, the topology of Lie groups and H-spaces, as well as many important questions from equivariant topology. In manifold topology we shall focus on some open problems related to new polynomial knot and link invariants in dimension 3, and to the study of the Seiberg-Witten invariants in dimension 4. We intend to use results from theoretical physics to obtain important new results concerning knots and links, in particular the magnetic field theory. In the piece-wise linear 3-dimensional topology we shall continue to study triangulations of Haken 3-manifolds and certain combinatorial moves - like the Reidemeister and the Pachner moves - which are related to some very deep problems in this area. In the infinite-dimensional topology we intend to continue our very successful studies of the Banach-Mazur compacta, using the new techniques we have developed in the past five years - and applied them to the solution of the celebrated West problem - to attack some deep questions related to the geometric structures of these spaces, which are of great interest also to convex geometers.
Significance for science
The research problems studied in this program have been in the center of attention of many experts in topology and geometry for a long time. The success in this area received much interest from the mathematical community. Our research will have a positive influence on further intense development of Slovenian mathematics, in particular topology and geometry and their connections to the international research network, especially within the European Union. Our research group is well established in its area and has received several domestic and foreign awards. Our results will be published in excellent international journals and the members of our group have received many invitations to important international conferences, which is an acknowledgement of the international recognition of our group. We registered an increased interest of foreign research institution for cooperation with IMFM, especially from the European Union. At present our research group has the largest number of international projects among all Slovenian research groups in the area of mathematics. We estimate that geometric topology and differential topology are two areas of basic research having the greatest potential for broad affirmation in the international scientific community. Furthermore, in recent years we successfully discovered new ways to apply our research results, e.g. we found an important application in chemistry and biology in the studies of the structure of DNA. We have discovered recently an innovative application of the knot theory in the magnetic fields theory. We also investigated fractal geometry which has wide applications.
Significance for the country
The research in the areas in this five-years program, had a very positive effect on the development of postgraduate studies in Slovenia in every university having PhD studies in mathematics. This is especially true for education of the collaborators of our program group at the Universities of Ljubljana, Maribor, Nova Gorica and of Primorska. Under the mentorship of our researchers and distinguished foreign researchers, our young researchers prepared their PhD’s in the most up-to-date topics of modern topology and geometry. Beside this we intensively cooperated with users, e.g. we prepared a new up-to-date course »Topology in computer science« at the Faculty of Computer Science and Informatics of the University of Ljubljana, which was of great interest also to experts in other fields, especially medicine. The algorithms which were developed for generating a discrete Morse function in computational topology can be successfully utilized in radiological diagnostics, e.g. CT, scintigraphy, internal medicine and urology. In this area we closely cooperated with industry with some cutting-edge domestic hi-tech companies.
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