Computationally intensive methods in theoretical computer science, discrete mathematics, combinatorial optimization, and numerical analysis and algebra with applications in natural and social sciences

In mathematical modeling, scientific computing, and data analysis, increasingly large amounts of data as well as increasing demand for computation capabilities are encountered. Addressing such challenges requires research and development of new methods and algorithms, in such a way that they can be reasonably implemented and used to tackle large problems on today's computers. Our research activities and goals will cover a selection of topics which revolve around computationally intensive methods for solving problems, formulating hypotheses, and performing analyses in a variety of domains, ranging from pure mathematics, to natural and social sciences. Our methodology combines the insights provided by deep mathematical and scientific results with their applicability in real-world situations. We will continue to develop our own software, as well as use and adapt existing technologies to tackle computationally demanding tasks. Our research group, which consists of more than 20 professional researchers, will focus on the following areas: Representations of graphs, maps, combinatorial configurations and other incidence structures. We will continue research of geometric, topological and combinatorial representations of graphs and combinatorial structures (configurations, maps, incidence structures, etc.). This involves structure analysis and classification of special families of objects. Databases of highly symmetrical combinatorial objects. We plan to build and later upgrade data collection of symmetric objects (symmetric graphs, maps, abstract polytopes, configurations) which will be used for hypotheses forming and testing. It will be made freely available to scientific community. Big data and network analysis. We will focus on development of a new approach to network analysis which is based on time quantities over semi-rings. We plan to work on the approach both from theoretical and algorithmical point of view. Theoretical computer science. We will study computational effects based on theory of computational effects built on algebra. The research topic will form a tight connection with programming and proof assistants. Numerical analysis and algebra. We will study systems of bivariate polynomials. We will study polynomial and rational parametric curves, surfaces and splines in the context of computer aided geometric design. Combinatorics. K-Schur functions will be studied. Various counting and enumeration problems will be addressed. Polytopes will be studied in the connection to the Tutte polytope. Applications in chemistry, synthetic biology and industrial applications. We will continue the already established cooperation in the fields of chemical graph theory and synthetic biology. We will be actively involved in cooperation with industry and seek for applications of knowledge in various business domains.

Results of our work in the previous research period show a tight interconnection of our research group with worldwide research. We publish our papers in the most influential scientific journals in many research areas, participate in the most important specialized international conferences, visit important foreign universities and scientific institutes, collaborate in international research and applicative projects. Many of our junior researchers have done at least part of their study at foreign universities of high quality. Our work is influential in the international research world and has plenty citations. We have an active collaboration with researchers from USA (Carnegie Mellon University, University of Pittsburgh, University of Pennsylvania, University of California Irvine, Temple University, Syracuse University, Colgate University, Drake University, Northeastern University), Canada (University of Waterloo, Simon Fraser University, York University), France (Universite de Marne-la-Vallee, INRIA Rocquencourt, INRIA Sophia-Antipoli, Universite Bordeaux I), Austria (Montanuiversitaet Leoben, RISC Linz, University of Vienna, FAS research Vienna), Germany (FU Berlin, Universität Bielefeld, TU Darmstadt, Universität Lepzig), Russia (Moscow state university), Italy (University of Pisa, University of Udine), Norway (CMA, IFI, University of Oslo, University of Trondheim), Netherlands (University of Eindhoven), Croatia (Institute Rudjer Bošković, University of Zagreb, University of Split), United Kingdom (University of Edinburgh, University of Birmingham, University of Sheffield), New Zealand (University of Auckland), Australia (University of Perth), Belgium (University of Ghent), Mexico (Universidad Nacional Autónoma de México), etc. We work on various interdisciplinary projects: with a group for analyzing social networks at FDV, Ljubljana, group for computer vision at FRI, Ljubljana, Institute of biophysics at MF, Ljubljana and National Institute of Chemistry. Members of our research group collaborate in European research projects and in various bilateral research projects (with USA, Italy, Norway, Croatia, Belgium) and in various applied projects. Our work yields important results in applications of computationally intensive methods in theoretical computer science, analysis of large networks symbolic computations, problems in graph theory, numerical analysis and linear algebra, computer aided geometric design, combinatorial optimization, chemistry and bioinformatics. The results of this research will prove very useful for all researchers in the area of algebraic combinatorics. Potentially, they can be used in either disproving or providing further evidence for long standing conjectures. The obtained catalogs can also be used to discover previously unobserved phenomena of the objects in question and thus inspire new direction of research. Symmetric functions play an important role in statistics and theoretical physics, k-Schur functions are also important in the theory of Schubert polynomials, Macdonald polynomials, cohomology etc. Polytopes are important in linear programming, statistics, graph theory and elsewhere. Many results have not only theoretical value, but can be applied in practice. We use this in transfer of knowledge from the scientific world into industry. For users outside scientific community the methodology of problem solving, mathematical modeling and design of computer algorithms and their implementation are of particular importance. We take care of applicative value of the obtained knowledge through cooperation with industry and knowledge transfer. Also the human resources and their development are important. We enable our junior researchers connections with leading international researchers and teach them to become an important part of the scientific community. Our results can be easily empirically verified by the number of our papers and citations of our work.

New algorithms and their implementations in symbolic computation and discrete and numerical mathematics enhance the efficiency and computing power of software tools in almost all areas of science and technology. The main impact of our research program on Slovenia can be divided into two groups: The development of the human potential: Our young researchers get the knowledge of the state-of-the-art in their respective fields. Some of them study abroad and work with the leading international authorities. All this will intensify international scientific cooperation and will help ensure a solid foundation and continued high standards of achievement in Slovenian science in the future. Applications and cooperation with potential users outside of the research community: Several members of our research programme collaborate in different applied projects in which they use in concrete situations the knowledge obtained in their fundamental research in the programme. Some of our researchers continue their careers outside of academia and transfer their acquired knowledge and skills (especially the problem-solving methodologies, but also their programming skills) to their working environment, thus intensifying the bonds between academia and the rest of society. Examples of our collaboration with different firms and institutions are Ambient d.o.o., Petrol d.d., Nigrad d.d., Abelium d.o.o., Xlab d.o.o., Luka Koper d.d., etc. Group members have contributed new approaches to the analysis of large networks that can be directly employed in public administration and in other areas. Our aimed projects for the Ministry of Defense and Slovenian army deal mostly with logistic support and e-learning. Our projects in the area of e-learning will simplify the preparation of integral learning programs incorporating information-communication technologies and media and will thus contribute to their wider accessibility and also to improvement of computer literacy in general. Our group is one of rare mathematical groups in Slovenia which has its own spin-off company, Abelium, established in 2009 by young PhDs coming form the research group. The company Abelium employs several PhDs and is becoming a partner in this research programme. The know-how developed in our research programme is nowadays highly appreciated in global economy (eg. network analysis, big-data, logistic optimizations). Since the Slovenian economy is involved in international activities it is extremely important that such knowledge is available to our companies, enhancing a faster and a more successful development.