Projects / Programmes
Geometry and topology of 3-manifolds
| Code |
Science |
Field |
Subfield |
| 1.01.02 |
Natural sciences and mathematics |
Mathematics |
Topology |
| Code |
Science |
Field |
| P150 |
Natural sciences and mathematics |
Geometry, algebraic topology |
| Code |
Science |
Field |
| 1.01 |
Natural Sciences |
Mathematics |
hyperbolic 3-manifold, geometric structure, Coxeter polyhedron, fundamental group, volume, rigidity, complexity, knot, link, branched covering, lens space, cyclicity
Researchers (17)
Organisations (1)
Abstract
The theory of 3-manifolds is a classical part of topology and geometry started by Klein and Poincaré. It has got a new life after the celebrated Thurston geometrization conjecture. In the last 20 years 3-manifolds admitting hyperbolic geometric structure (as well as hyperbolic 3-orbifolds and cone-manifolds) have been intensively studied. In the present project we shall develop theory of 3-manifolds in several directions, concentrating on two key characterizations of manifolds.
In the course of investigating fundamental polyhedra of hyperbolic 3-manifolds we shall describe properties of their volumes, which are topological invariants by the Mostow rigidity theorem. By the classical Alexander theorem, any closed 3-manifold is a branched covering of the 3-sphere. We shall develop a universal approach to constructing closed 3-manifolds which are cyclic branched coverings of lens spaces, and to studying their topological properties.
Both characterizations will be applied to estimate the complexity (in the sense of Matveev) of obtained 3-manifolds, which is a very useful parameter for tabulating 3-manifolds. Moreover, both characterizaions will find applications in combinatorial group theory: the polyhedral construction will be related to the studies of torsion-free subgroups of Coxeter groups and the cyclic covering construction will be related to the investigations of cyclically presented groups – their finiteness and hyperbolicity.
Significance for science
This research project treated one of the most active parts of modern mathematics: in recent decades low-dimensional topology has witnessed an explosive growth and its researchers have been awarded some of the most prestigeous prizes (notably Perel'man for his proof of the over hundred years old Poincaré Conjecture). This is therefore an area in which very important results follow faster than in many other areas, also because of the many strong relations with other areas of mathematics (notably graph theory, functional analysis, algebraic geometry, and quantum field theory). This project concerned some key important subjects and we found new methods and techniques for resolving very difficult open problems which have been in the center of attention by many leading experts from topology and geometry for a long time. Therefore our results will definitely have impact on the mathematics and will also accelerate the progress of mathematics in Slovenia. We also discovered new ways to apply our results, e.g. our work had an important application in chemistry and biology in the studies of the structure of DNA. We also found new applications of knot theory in magnetic field theory and investigated fractal geometry which has broad applications. Our results received a lot of interest from the international mathematical community. Group members published extensively in international mathematical journals placed high of the SCI list (e.g. Advances in Mathematics, Mathematische Annalen, Mathematical Research Letters, Proceedings of the Royal Society, Transactions of the American Mathematical Society, etc.). Some of our publications with the Elsevier North-Holland publishers were among the most downloaded (e.g. in Topology and Its Applications). Our group is well established in its area and it has received many domestic and foreign awards in the past. Members of our group received invitations to give lectures at important international conferences, confirming the international recognition of our research group. We had increased interest of foreign research institution for cooperation with our institute, especially from European Union. As a result, our research group has the largest number of international project in mathematics. The project also had a very positive influence on development of Slovenian mathematical school, emphasizing topology and geometry, and connection to the research networks, especially in the European Union.
Significance for the country
The main result of this project is a discovery of very important new fundamental laws and their applications in mathematics, and extending the available research tools in modern geometric topology to dimension 3, 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 geometric topology of low dimensions and one of the important meeting points of experts in 3-manifolds. Our research 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 cooperated 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 project had an extraordinary positive effect on the development of graduate studies in Slovenia, in particular 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 theses on the most up-to-date topics in topology and geometry. We offered modern graduate course like »Topology in computer science« at the Faculty of Computer Science and Informatics at the University of Ljubljana, which was of interest also for other fields, especially medicine.
Most important scientific results
Annual report
2011,
2012,
2013,
final report,
complete report on dLib.si
Most important socioeconomically and culturally relevant results
Annual report
2011,
2012,
2013,
final report,
complete report on dLib.si
