Projects / Programmes
January 1, 2009
- December 31, 2014
| Code |
Science |
Field |
Subfield |
| 1.01.00 |
Natural sciences and mathematics |
Mathematics |
|
| Code |
Science |
Field |
| P150 |
Natural sciences and mathematics |
Geometry, algebraic topology |
| Code |
Science |
Field |
| 1.01 |
Natural Sciences |
Mathematics |
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 (25)
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 were in the center of attention of many experts in topology and geometry for a long time. Our success in this area therefore received great interest from the mathematical community. Our research had a positive influence on further intensive development of Slovenian mathematics, in particular topology and geometry and their connections with the international research network, especially within the European Union. Our research group is well established in the area and it has received several domestic and foreign awards. Our results were published in excellent international journals and the members of our group 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. Our research group has the largest number of international projects among all Slovenian research groups in the area of mathematics. Geometric topology and coarse geometry are the two areas of mathematics which have the greatest potential for broad affirmation in the international scientific community. We discovered new ways to apply our results, e.g. we found an important application in chemistry and biology in the studies of the structure of DNA. We discovered an innovative application of the knot theory to the magnetic fields theory. We also successfully investigated in fractal geometry which has wide applications.
Significance for the country
The main result of this program was a discovery of very important new fundamental laws and their applications in mathematics, and extending the available research tools in modern topology and geometry, and its applications, as well as further development of research in mathematics in Slovenia. Our results agree with plans for the development of Slovenian science and technology, in the field of enhancement of knowledge, i.e. on the progress of science as well as on the substantial improvement of the quality of the doctoral program. Our research is related to and builds upon past successful research in this field and is connected with the problems which have been very successfully studied with very positive feedback in numerous international projects of our research. As the result of our longstanding efforts our institute is an internationally renowned European center of topology. Our group has received several national and international prizes and we were selected among the best program teams in the country. Several members of the group are already very influential in international research in their fields of expertise. Our younger researchers, working within our group, have very successfully began to establish themselves. In future we shall significantly expand our work on applied aspects of topology and strengthen our position in the EU research network. We successfully cooperate with the industry, e.g. we developed new and effective algorithms for generating discrete Morse functions in computational topology, which can be applied in radiological diagnostics, e.g. in CT, scintigraphy, internal medicine and urology. In this areas we are cooperating with some cutting-edge domestic hi-tech companies. Therefore we plan such productive collaboration also in the future. The program had an extraordinary positive effect on the development of graduate studies in Slovenia, in particular of the PhD programs in mathematics at the University of Ljubljana. Under the mentorship of our researchers and leading foreign researchers, our young researchers prepared their dissertations on the most up-to-date topics in topology and geometry. We offered several modern graduate courses, e.g. »Topology in computer science« at the Faculty of Computer Science and Informatics at the University of Ljubljana, which are of interest also to experts from other fields, especially medicine.
Most important scientific results
Annual report
2009,
2010,
2011,
2012,
2013,
final report,
complete report on dLib.si
Most important socioeconomically and culturally relevant results
Annual report
2009,
2010,
2011,
2012,
2013,
final report,
complete report on dLib.si
