Algebra, discrete mathematics, probability and game theory

The research program is a natural follow up of the research program P1-0285 2015-2021. It is of intra- and inter-disciplinary nature. Interactions of combinatorics and algebra with other mathematical disciplines as well as applications in other sciences, mathematical biology, probability and game theory are studied. Reseach content and goals of this research program are divided into six fields: (1) Algebra with Combinatorics and Graph Theory, (2) Other Algebraic Tools for Studying Graph Properties, (3) Hereditary Graph Classes, (4) Mathematical Studies of the Evolutionary Dynamics, (5) Probability, (6) Game Theory. Combinatorial language can often provide useful insights when studying structural properties of certain algebraic objects such as permutation groups. Conversely, when investigating structural properties of combinatorial objects admitting transitive group actions (or possibly intransitive actions with a small number of orbits), one often relies on purely algebraic results. An example of such a fruitful interplay is the main topic of the program (covering (1) - (3)), which draws together open questions from combinatorics and algebra: groups acting on graphs, structural properties of graphs with prescribed symmetry, graph covers, blocks of imprimitivity, even/odd automorphisms, intersection densities, Schur rings, Terwilliger algebras, preservers, etc. The Classification of finite simple groups (CFSG), while having proved to be a powerful tool in this kind of research, may sometimes obscure the structural content of the problems dealt with. Consequently, searching for direct CFSG-free proofs, whenever possible, will also be a focus of P1-0285. The proposed program also aims at making UP FAMNIT (the establishment of which has been an important achievement of P1-0285) a Center of Excellence in mathematics, modeled after some existing centers in certain countries comparable to Slovenia. As a consequence of ensuing strengthening of mathematical research at the University of Primorska and the two partner institutions (University of Nova Gorica and Institute of Mathematics, Physics and Mechanics), a spillover effect is expected in Slovenian science and beyond.

Apart from Art, Mathematics is the only universal language of human communication present in all developed civilizations. It is a foundation of Natural Sciences and lies at the heart of theoretical Computer Science. Abstract mathematical theories are used in Physics, Engineering, Computer Science and also in Social, Economic and Biomedical sciences. It is therefore to be expected that Mathematics will influence several important areas of human activities, which will have a crucial impact in the 21st century, such as modern Economy, Psychology, Sociology, Medicine, Safe Communications, Data protection, Decoding of Humane Genome. The program stands at the cutting edge of today's research in all of the proposed areas. The obtained results will certainly help to better understand the subtle connections between Algebra and Combinatorics, and Graph Theory in particular. Structural results about objects, measured by the degree of their symmetry, are important in general, not only in mathematics but also in other sciences; for example in chemistry and in the theory of elementary particles. Recently, graphical models have become a central paradigm for knowledge presentation and reasoning. They are at the heart of research in many statistical, computational, and mathematical fields, including bioinformatics, communication theory, combinatorial optimization, computer vision, signal processing, information theory, coding theory, information retrieval, and statistical machine learning. Due to current situation regarding Covid-19 let us point out the importance of our research work within mathematical biology. During the ongoing Covid-19 pandemic it has become clear that mathematical models play an important role in understanding and controling the spread of the virus. In addition to the very specific mathematical and statistical models that are used on a daily basis to predict the spread and to assess the effectiveness of different control measures it is also important to develop more conceptual models that allow us to understand how population structure in interactions between different strains of the pathogen and the hosts shape the epidemiological and the evolutionary dynamics. We expect our studies of the eco-evolutionary dynamics of respiratory infections to represent important contributions to science. The importance of our research goals is evident from the bibliographies of the group members for the period 2015-2021, from the citations of obtained results, numerous links with foreign scientists and comprehensive world bibliography in general. It is to be expected, that just as with the results from the 2015-2021 period, the results obtained during the next program period will be published in prestigious international journals and presented at international conferences. As a consequently, the Slovenian mathematical school in Algebra, Combinatorics, Graph Theory, Probability, Mathematical Biology and Game Theory will gain further worldwide recognition.

The proposed research program will help Slovenia to stay in touch with current trends in mathematics. Moreover, the inclusion of Game Theory goes hand in hand with contemporary methods of mathematical modeling based on computer simulations and laboratory verifications. But perhaps the most important part of this program is to support further improvement of the School of Mathematics at UP FAMNIT (established by this group in 2007), which has already gained international recognition due to its outstanding scientific achievements and has become one of the best meeting points for international scholars and students. The times of social changes call for further inclusion of Mathematics into both research and educational curriculum, thus enabling a faster technological development in Slovenia. The research program P1-0285 comprises various parts of Applied Mathematics and has a clear interdisciplinary flavor, and so the expected scientific results of our research are and will continue to be widely useful in different high-tech industrial sectors such as IT sector, in economical, financial and insurance sectors, army and military industry. Also, our existence as a fully developed nation in Europe depends as much on preserving our language and culture as it depends on having a highly educated population. It is thus necessary to be able to use different communication channels - and mathematics, as a universal language, is a key factor here. Hence, the above discussion shows that the P1-0285 program plays an essential role in the strengthening of Slovene national identity, in particular in its western, most vulnerable part of Primorska. The P1-0285 program will make important contribution to achieving national goals indicated in the Research and Innovation Strategy of Slovenia 2021-2030 and in the National Higher Education Strategy of Slovenia 2021-2030 (both documents are currently in the final stage of adoption and the P1-0285 members are actively involved in the preparation). Also, the P1-0285 research work plan for the forthcoming funding period is in line with EU strategic plan 2021-2024 adopted by the European Commission for Horizon Europe.