Projects / Programmes
Topology and its applications
January 1, 2022
- December 31, 2027
| Code |
Science |
Field |
Subfield |
| 1.01.00 |
Natural sciences and mathematics |
Mathematics |
|
| Code |
Science |
Field |
| 1.01 |
Natural Sciences |
Mathematics |
knot theory, knot invariant, biomolecule knotting, topological complexity, robot motion and manipulation, minimal triangulation, random flag complex, persistent homology, toric topology
Data for the last 5 years (citations for the last 10 years) on
June 28, 2024;
A3 for period
2018-2022
| Database |
Linked records |
Citations |
Pure citations |
Average pure citations |
| WoS |
555 |
4,179 |
3,245 |
5.85 |
| Scopus |
563 |
4,394 |
3,426 |
6.09 |
Researchers (12)
Organisations (3)
Abstract
The proposed 6-year national research program P1-0292 "Topology and its Applications (2022-2027)" is a continuation of our 7-year national research program P1-0292 "Topology, Geometry and Nonlinear Analysis (2015-2021)". In the next 6-year period (2022-2027), we shall focus on several unsolved problems in modern topology and its applications in other fields of science and technology. Topology has evolved from an almost exclusively theoretical subject to a branch of contemporary mathematics where application questions are becoming an increasingly important motivation for the development of the theory. At the same time, new methods for solutions of a wide range of problems are emerging and evolving, ranging from pattern and shape recognition, noise control, sensor networks, topological data analysis, and material science, to genomics, evolutionary biology, neural science, robot motion and manipulation planning, CT and scintigraphy analysis, and many more. Our research group has already actively participated in the past in several different applied areas, e.g. in studies of neural structures (Blue Brain Project), DNA knotting, topological analysis of CT and scintigraphy data, complexity analysis of robot motion, and manipulation algorithms. It is our plan to continue in 2022-2027 along this path and to combine theory with development of new applications of algebraic and geometric topology. Our main research goals in 2022-2027 will be: (i) to extend the notion of knotted structures and develop invariants and classification methods for higher-dimensional knots, (ii) to apply machine learning to identify knotting patterns of biomolecules, (iii) to study complexity of robot motion under varying ambient conditions, (iv) to study minimal triangulations of manifolds, (v) to develop new types of random flag complexes to model neural networks, and (vi) to relate persistent homology to geometric features of the underlying space, such as sets of geodesics and thick-thin decompositions. We shall continue in 2022-2027 to publish our results in excellent journals in pure and applied mathematics. We shall also continue to collaborate with leading research groups from European Union, United States, Russian Federation, and Japan, in the framework of international (bilateral and multilateral) research projects and networks. We shall continue organizing conferences and workshops, with the participation of key foreign experts, thus enabling intensive exchange of knowledge and further collaboration. We shall also continue developing doctoral program in topology and its applications in Slovenia and including doctoral students in our research.
Significance for science
The proposed program for the period 2022-2027 is on very current research areas of modern topology and its applications, in particular rapidly developing set of problems in the area that is usually denoted by the acronym ACAT - Applied and Computational Algebraic Topology. These questions are of great interest to experts in both pure and applied mathematics from around the world (European Union, Russian Federation, United States, Japan, and elsewhere). Our program group has been very successful in solving important problems of topology and geometry since the 1980s, and in the last funding period (2015-2021) we have also included an important area of nonlinear analysis in our work. Some of the proposed problems have been asked or conjectured by leading researchers from around the world. Therefore, our expected results will attract great interest not only in the international mathematical community, but also in other sciences, where they are increasingly resorting to methods derived from algebraic and geometric topology. The most notable novelty in 2022-2027, compared to the previous period, is the study of problems that have a background in biological sciences. One important topic involves the study of knotting phenomena of biomolecules using methods from the theory of knots and links. We shall also study the connectivity structures that arise in complex networks, especially the nervous system. In recent years, researchers involved in a large interdisciplinary collaboration called the Blue Brain Project have found that already in certain organisms, the homological connectivity of the connectome (neuronal connectivity graph) deviates significantly from conventional models of randomized networks. This discovery has opened up a whole range of questions and the search for network models that could describe structures found in nature. We shall continue in 2022-2027 to publish our results in journals with a high SCI factor - in the previous period (2015-2021) we published over 200 articles in journals listed in the first quarter of the SCI list. Our program group has always had the most international projects among all research programs in mathematics in Slovenia, which witnesses to our deep involvement in international research networks. Another very important area of our work is authoring comprehensive monographs in the fields of our expertise, thus promoting their development as well as our group around the world. In the period 2015-2021 we published several monographs with leading scientific publishers (e.g. Springer, Cambridge, Francis & Taylor), including the first monograph on surgery on generalized manifolds (A. Cavicchioli, F. Hegenbarth, D. Repovš. Higher- Dimensional Generalized Manifolds: Surgery and Constructions, Zürich: European Mathematical Society, 2016. [COBISS.SI-ID 17667161]). Also in 2022-2027 we plan to continue publishing monographs. We shall also continue to disseminate our results also through international meetings - in the past we have successfully (co)organized several conferences in Slovenia and abroad (Austria, Canada, Czech Republic, Israel, Italy, France, Japan, Norway, Poland, Russian Federation, United States, and elsewhere). We shall continue to serve on editorial boards of excellent mathematical journals: Boundary Value Problems (Springer), Complex Variables and Elliptic Equations (Taylor & Francis), Fixed Point Theory and Algorithms for Sciences and Engineering (Springer), Journal of Mathematical Analysis and Applications (Academic Press), Mediterranean Journal of Mathematics (Birkhäuser), and others. Also in 2022-2027 we expect to receive invitations for plenary talks at international conferences and colloquia at leading universities and research institutions. Our researchers have already been elected to foreign academies (e.g. European Academy of Sciences and Arts) and awarded honorary doctorate and international prizes and medals. Therefore we expect more such recognition also in the future.
Significance for the country
I. Expected direct impact of our research in 2022-2027: Our program group has a long and successful history of cooperation with industry since the 1980's. Members of our group have so far collaborated on the following applied projects: (i) Telecommunications: with the national telephone company Iskra Telematika research laboratories on optimization of switchboards. (ii) Electrotechnics: with the company Elektrina on randomness verification and irregularity analysis of a mechanical random generator. (iii) Optimization of shapes: with the shipbuilding company Seaway group on optimization methods for shapes of yacht hulls. (iv) Pharmaceutical industry: with the main national pharmaceutical company Krka in randomized cancer studies. (v) Electronics: with company Elatec on algorithms for digital scanning probe microscopes controller with feedback for regulation of vertical moves. (vi) Optimization and logistics: with high-tech company Abelium and start-up company Epilog on integrated software solutions for internal logistics and automated warehouses. (vii) Fraud and anomalities detection: with company Optilab on technological solutions in the field of fraud and anomalies detection with efficient support. (viii) Textile industry: with the Faculty of Natural Sciences and Engineering, University of Ljubljana, on analyzing the influence of chemical finishes on the surface properties of glass plate. (ix) Laser technology: with the company Pangolin on development of control systems for laser projections. We shall continue our collaboration with the industry also in 2022-2027, applying our extensive expertise. The proposed topics are already well-known for their applicability, e.g. knots and links theory have found surprising applications in chemistry and biology, in particular for the DNA structure (in the past we have discovered also applications of knots in magnetic field theory). We shall also work in fractal geometry which has many applications. Next, our algorithms which we developed for generating a discrete Morse function in computational topology, can be successfully used in radiological diagnostics, e.g. in computer tomography imaging and scintigraphy in medicine (we already closely collaborate with some cutting-edge high-tech companies). II. Expected indirect impact of our research in 2022-2027: We emphasize the key 3 impact areas: (A) Promotion of Slovenia: Our group will continue to promote Slovenian science via most effective dissemination of our research results: (i) by publishing our papers in excellent mathematical SCI journals, (ii) by publishing our monographs with leading publishers, (iii) at invited plenary lectures at international conferences and workshops, (iv) at colloquia at leading foreign research institutions, (v) participating in international research projects, (vi) by having distinguished scientists visit Slovenia and work with our group, (vii) via national and international awards and prizes for our work. (B) Education in Slovenia: Our research group will also continue to pay great attention to development of the (undergraduate and graduate) mathematics education in Slovenia: (i) Developing university curriculum by offering modern courses at the University of Ljubljana (and writing textbooks) on topics in topology and its applications, e.g. for graduate courses Topology in computer science, Computational Topology, and Mathematical modelling at the Faculty of Mathematics and Physics, and the Faculty of Computer Science and Informatics (these courses are of great interest also to experts in other fields). (ii) Due to our excellent international collaboration, we shall have numerous visiting professors from all over the world giving courses to our doctoral students, which are always on the forefront of the mathematical research. Many will spend extended period of time in Slovenia and can be coadvisers to our Ph.D. students. (iii) Organizing schools and workshops in Slovenia with participation of leading experts, e.g., in the framework of the European Science Foundation research network Applied and Computational Algebraic Topology, we organized in 2013 a school on computational topology and topological data analysis, with lecturers from EU and USA, attended by over 60 PhD students from 16 countries. (iv) Enabling our postdocs to continue training abroad, sending them to leading universities in EU and USA, e.g. our young researcher Sara Kališnik continued at Stanford University. We shall also continue providing for outstanding foreign students to study in Slovenia for a PhD within our research group. (C) Slovenian culture: a very important work that our research group has done in the last 40 years is for terminology: we have been systematically preparing the English-Slovenian mathematical dictionary, with emphasis on the areas of our expertise, and we have included also many other areas. We also plan to prolong in 2022-2027 our strongest efforts for the optimal impact of our research on Slovenia's socio-economic and cultural development, as we have very successfully done in the past.
