Projects / Programmes
Topology, geometry and nonlinear analysis
January 1, 2015
- December 31, 2021
| Code |
Science |
Field |
Subfield |
| 1.01.00 |
Natural sciences and mathematics |
Mathematics |
|
| Code |
Science |
Field |
| P001 |
Natural sciences and mathematics |
Mathematics |
| Code |
Science |
Field |
| 1.01 |
Natural Sciences |
Mathematics |
geometric topology, algebraic topology, topology of low-dimensional manifolds, differential geometry, coarse geometry, fractals theory, applied and computational topology, nonlinear elliptic partial differential equations, applied nonlinear analysis
Researchers (22)
Organisations (1)
Abstract
The proposed national research program Topology, geometry and nonlinear analysis is a continuation and a significant extension of our current 6-year national research program P1-0292-Topology and geometry (which was chosen by the Slovenian Research Agency among the best Slovenian research programs). We plan to continue working on several outstanding unsolved problems in geometric topology, algebraic topology, topology of low dimensional manifolds, differential geometry, coarse geometry, fractals theory, and applied and computational topology. We shall also significantly expand our research in nonlinear analysis and its applications, where we have already achieved several excellent results in recent years.
In the next period we plan to concentrate on the following research topics: (i) inverse limit spaces of sequences with multivalued bonding maps, (ii) 3-manifold properties related to their end structures, (iii) iterative systems that embed certain limit sets in the Euclidean 3-space, (iv) algebraic structure of the embedding homeomorphism groups of 3-manifolds, (v) algebraic topology of wild spaces, in particular grope group theory, (vi) fundamental groups of compacta and lifting spaces, (vii) homotopy theory methods in fibrewise topology, (viii) application of topological methods in complex analysis, in particular investigations of Stein manifolds, (ix) new approach to persistent homology, (x) open problems in applied topology, (xi) investigating asymptotic properties of metric spaces by means of coarse geometry, (xii) nonlocal equations involving a general integro-differential operator of fractional type, (xiii) elliptic Kirchhoff-type problems concerning evolution phenomena with an intrinsic nonlocal nature, (xiv) parametric differential inclusion problems involving discrete operators in presence of different boundary value conditions, or requiring suitable conditions on the nonlinear term in presence of critical growth, and (xv) critical and singular problems with variable exponent in the qualitative analysis of nonlinear elliptic PDE’s.
As we have successfully done in the past, we plan to publish all our results in excellent journals for pure and applied mathematics. We also have plans with leading scientific publishers for several new monographs. Our investigations will continue to be done in intensive collaboration with leading research groups from the European Union, the United States, Canada, Russian Federation, China and Japan, mostly in the framework of the current (and future) international (bilateral and multilateral) research projects and networks. We also plan several conferences and workshops during this period, with participation of the key foreign experts, so that new results can be presented and intensive exchange of expertise is enabled. We plan to continue with our successful applications of our results outside mathematics. We shall further develop the PhD program in these research areas in Slovenia and shall intensively include PhD students in our research.
Significance for science
This program proposal deals with very current research topics in modern topology, geometry and nonlinear analysis which are of substantial interest to experts in both pure and applied mathematics around the world (European Union, Russian Federation, United States, Canada, China, Japan and elsewhere). Our program group has successfully been doing research on outstanding problems in topology and geometry since the 1980’s, and has in recent years expanded to include the important area of nonlinear analysis with applications in the most recent financing period (2009-2014).
Several of the currently proposed problems have been around for a long time and many excellent researchers around the world have already attempted to work on them. Therefore our expected results on all these topics will be of great interest to the international mathematical community and an important contribution to the knowledge in these areas of pure and applied mathematics. Our group's excellence has been internationally recognized in several areas of our research, notably in geometric topology and nonlinear analysis, proven by citations, invitations and awards we have received.
Besides its important international scientific impact, our new six-year program proposal is built to have a very important extra goal – it will strongly develop in Slovenia (especially, if we get the additional program research funding, cf. item 23) one of the most important applied research areas of mathematics, namely nonlinear (ordinary or partial) differential equations and applied functional analysis. Nonlinear analysis is no doubt one of the areas of the fundamental research which has the greatest potential for a strong impact and consequently wide affirmation in international scientific community.
Differential operators, which describe the behavior of fluids are by rule nonlinear and present very difficult mathematical problems. The behavior of the spectra of these operators is so unusual that their description represents a substantial expansion of our knowledge - in the area of mathematical analysis as well as in the area of physics of fluids. The research problems which we propose to study have been inspired by some very concrete models arising in mathematical physics and they have already been studied by several well-known experts abroad. Our program is very strong and it heavily relies on mathematical knowledge from several research fields, including nonlinear functional analysis, differential and partial differential equations, geometry, topology, and the calculus of variations. Because of its potential for applications and due to expected important impact of planned results on the entire area of theory of partial differential equations, in which there are still open fundamental problems, we believe that this program proposal should have a very high priority, due to its relevance and importance, both in Slovenia as well as abroad.
We shall continue to submit our results to journals with high SCI impact factor – in the previous period (2009-2014) we have published 53 papers in journals in the top quarter of the SCI list - the list was attached to the program report for this period. Our program group has always had by far the greatest number of international projects among all mathematics research groups in Slovenia, which attests to our excellent embedding into international research network. We have had 153 foreign coauthors in 2009-2014 period (several of which have visited Ljubljana and are listed in item 22 below) from Australia, Austria, Belarus, Bulgaria, Canada, Cina, Czech Republic, France, Germany, Georgia, Greece, India, Iran, Ireland, Israel, Italy, Japan, Korea, Morocco, Poland, Romania, Russian Federation, Saudi Arabia, Spain, Tajikistan, Tunisia, Ukraine, and United States.
Another very important part of our work is writing comprehensive monographs on the areas of our expertise, by means of which we promote the development of the areas as well as p
Significance for the country
I.Direct impact: 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 performance of digital telephone switchboards.
(ii)Electrotechnics: with the company Elektrina on randomness verification and irregularity analysis of a mechanical random generator.
(iii)Optimization of shapes: with the ship building company Seaway group on optimization methods for shapes of yacht hulls.
(iv)Process optimization: with companies Abelium and Epilog on devising new technological solutions in optimization of the logistics of large storage capacities.
(v)Pharmaceutical industry: with the main national pharmaceutical company Krka on various statistical analyses.
(vi)Electronics: with company Elatec on algorithms for digital scanning probe microscopes controller with feedback for regulation of vertical moves.
(vii)Optimization and logistics: with high-tech company Abelium and start-up company Epilog on integrated software solutions for internal logistics and automated warehouses.
(viii)Fraud and anomalities detection: with company Optilab on technological solutions in the field of fraud and anomalies detection with efficient support.
(ix)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.
We shall continue our collaboration with industry, applying our expertise in topology, geometry and nonlinear analysis. These areas of mathematics are 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 (and we have discovered another application of knots in magnetic field theory). We are working on fractal geometry which has many applications. Next, our algorithms which were developed for generating a discrete Morse function in computational topology can be successfully utilized in radiological diagnostics, e.g. computer tomography imaging and scintigraphy in medicine (in this field we already closely cooperate with some cutting-edge high-tech companies). Numerous applications of nonlinear analysis are known and many of them are studied in our monograph Variational Principles in Mathematical Physics, Geometry, and Economics, Cambridge University Press 2010.
II.Indirect impact: We emphasize the most important impact factors in this respect:
(A) Promotion of Slovenia: Our group has always made best efforts to promote Slovenian science via most effective dissemination of our research results:
(i)publishing our papers in excellent mathematical SCI journals
(ii)publishing our monographs with leading publishers
(iii)invited plenary lectures at international conferences and workshops
(iv)colloquia at leading foreign research institutions
(v)participating in international research projects
(vi)having distinguished scientists visit Slovenia and work with our group
(vii)awards and prizes which our members have received for their scientific work
(B) Education in Slovenia: Our research group has always paid particular attention to development of the (undergraduate and graduate) mathematics education in Slovenia and we shall continue these efforts also in the next financing period of this program:
(i)Developing university curriculum by offering modern courses at the University of Ljubljana (and writing textbooks) on topics from topology, geometry, nonlinear analysis and applications, e.g. 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 for e
Most important scientific results
Annual report
2015,
interim report
Most important socioeconomically and culturally relevant results
Annual report
2015,
interim report
