Projects / Programmes
Limiting sets of iterative systems in dimensions 3 and 4
| 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 |
Limit set, iterative method, 3-manifold, Cantor set, rigid embedding, 3-dimensional Euclidean space, genus, fundamental group, classification of embeddings, link, application of topology, dynamical system, chaos, fractal geometry, Hausdorff dimension
Researchers (16)
Organisations (2)
Abstract
This is a proposal for a 3-year research project, by means of which we wish to continue and expand (in intense collaboration with some leading research groups from the European Union, the Russian Federation, the United States and Japan) our international prominence in our use of iterative topological and geometric techniques with a goal of producing limit sets, three manifolds, algebraic examples, and dynamical systems with specific properties.
Our primary goal is to gain a better understanding of topological, dynamical and algebraic properties of limit sets of iterative systems. There is a close relationship between the properties these sets, the geometric techniques used in their production, and the algebraic structure of the limit homeomorphism groups of the end structure of the complement.
The main theme of the proposed research is the close interdependence of 3-manifold properties related to the structure at infinity (ends), iterative systems that embed certain limit sets in 3-dimensional space, and the algebraic structure of the homeomorphism group of these embedded limit sets that come from homeomorphisms of the ambient space. These homeomorphisms are also the homeomorphisms of the end structure that extend. Better understanding of the types of the algebraic, topological, and geometric properties these limit sets have can have should lead to a better understanding of iterative, dynamical, and algebraic systems.
Our work on embedding homogeneity groups ties together these concepts and is likely to open up new research areas linking these various structures. Our results leads to a better understanding of the 3-manifold complements of these limit sets. The study the mapping class group structure of 3-manifolds is an extremely important area of current research. Our projects bring a new way of looking at the geometric, algebraic and dynamical structures.
Our research in the last ten years led to a much better understanding of how to distinguish between various types of limit sets and how to produce limits with certain specified topological and algebraic properties such as rigidity or types of homogeneity. Our techniques generalize to related constructions yielding Whitehead type links and manifolds. We produced breakthroughs in the types of examples with rigid or other algebraic structures. The continued use of our techniques will produce original results of widespread interest.
Our focus has been on examples that arise as nested intersections or unions of tori and related manifolds. The focus on intersections produced an example that was Lipschitz ambiently homogeneous. This depends on a particular Antoine Cantor set with geometric self-similarity. We then produced uncountably many inequivalent such examples, each with the same number of components at each stage. Rushing showed that there are Cantor sets in R3 of varying Hausdorff dimension.
We propose to find self-similar Cantor sets in dimensions ) 3 of each possible Hausdorff dimension. This is related to producing self-similar Blankinship Cantor sets in dimension 4. We produced examples that are rigidly embedded with simply connected complements, and we characterized such compacta arising from Bing-Whitehead constructions. A natural project is to extend these examples to higher dimensions.
Another main proposed area of work is to produce examples with specific embedding homogeneity groups related to the mapping class group of the complement. Our recent constructions could be used and varied to produce these examples.
Recently, Gabai has shown that the Whitehead 3-manifold can be written as a union of two copies of R3 with intersection R3. We will utilize our expertise in working with Cantor sets to investigate which contractible 3-manifolds that arise as increasing unions of tori can be written as such a union. We expect to characterize classes which can be decomposed this way and classes which cannot.
Significance for science
This research project investigated several key important topics in modern geometric topology. We have found new methods and techniques for resolving several open problems which have been in the center of attention by many leading experts for a long time. Therefore our results will definitely have impact on this field of mathematical sciences and will also support the progress of mathematical sciences in Slovenia. We have also discovered new ways to apply our results in other areas. We published our results in international mathematical journals which are placed high on the SCI journals list, e.g. Advances in Nonlinear Analysis, Annales de l'Institut Henri Poincare, Archive for Mathematical Logic, Communications in Contemporary Mathematics, Communications on Pure and Applied Analysis, Computers & Mathematics with Applications, Discrete and Continuous Dynamical Systems, Fractional Calculus & Applied Analysis, Geometry & Topology, Journal of Mathematical Analysis and Applications, Journal of Symbolic Logic, Journal of the London Mathematical Society, Linear Algebra and its Applications, Nonlinearity, Proceedings of the Royal Society Edinburgh, Results in Mathematics, Transactions of the American Mathematical Society, etc. Our results received a lot of interest from the international mathematical community and have been cited many times. Some of our publications with the Elsevier North-Holland publishers were also among the most downloaded (e.g. in Topology and Its Applications). The project group is already very well established in its research area and it has already received many domestic and foreign awards. Members of our group have received numerous invitations to give lectures at important international conferences, confirming the international recognition of our research group. We see an increased interest of foreign research institution for cooperation with our institute, especially from EU. As a result, our research group has many international projects.
Significance for the country
The work on this project will have a very positive influence on development of mathematical research in Slovenia, with the emphasis on topology and its applications, and on its connection to the research networks worldwide, in particular to the EU. The main results of this project is a discovery of very important new fundamental laws and their applications in mathematics, and extending the available research tools and their applications. Our results fit in very well with the plans for development of Slovenian science and technology, in the field of expansion of knowledge, as well as for a 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 a 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 and one of the important meeting points of experts in this area. Our research has received several national and international prizes and we have been selected among the best program teams in the country. Several members of the group are already very influential internationally in their field of expertise. Our younger researchers, working within our group, have also very successfully began to establish themselves. We are very successfully cooperating with other sectors, e.g. we have developed new effective algorithms for generating discrete Morse functions in computational topology, which can be applied in medical radiological diagnostics. In this areas we are cooperating with some cutting-edge domestic hi-tech companies. Therefore we plan such productive collaboration also in the future. We shall also shall significantly expand our work on applied aspects of topology and strengthen our position in the EU research network. The project had also 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 our mentorship, our young researchers prepared their theses on the most up-to-date topics in topology and its applications. We have prepared modern graduate courses, e.g. »Computational topology« at the Faculty of Computer Science and Informatics at the University of Ljubljana, which is of interest also to other fields, in particular in medicine.
Most important scientific results
Annual report
2014,
2015,
final report
Most important socioeconomically and culturally relevant results
Annual report
2014,
2015,
final report
