Head of program P1-0288: prof. dr. Primož Potočnik

Duration: 01/01/2020 - 12/31/2025

Abstract: In mathematical modeling, scientific computing, and data analysis, increasingly large quantities of data as well as increasing demand for computation capabilities are being encountered. 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 and industry. Our methodology combines the insights provided by deep mathematical and scientific results with their applicability in real-world situations. 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 into geometric, topological and combinatorial representations of graphs and combinatorial structures (configurations, maps, maniplexes, polytopes, etc.). This involves the structural analysis and classification of special families of objects. Databases of highly symmetrical combinatorial objects. We plan to build and upgrade data collections of symmetric objects (symmetric graphs, maps, abstract polytopes, configurations) which will be used for hypotheses forming and testing. Collections are freely available to the scientific community. In the process, the theoretical background will be studied and developed for specific collections. Big data and network analysis. We will focus on the development of new approaches to network analysis with a focus on linked networks, temporal networks and symbolic networks. We will further develop methodologies for symbolic data analysis on big data. The work on new approaches will address both theoretical and algorithmic points of view. Computational reasoning. We will work on two main topics. The first is the development of domain-specific methods for reasoning about computational systems. The second is the development of a unified approach to the implementation of configurable dependent type theories. The topics will be synthesized in applications of proof assistants and in the design of bespoke type theories designed around programming applications. Numerical analysis and algebra. We will study systems of bivariate polynomials and applications, new interpolation and approximation schemes, the inverse eigenvalue problem and cubic splines in the context of computer aided geometric design. Combinatorics. We will work on hyper-hypergeometric and convolutive solutions of linear recursions with hypergeometric coefficients, representations of the symmetric group related to parking functions, Schur functions and alternating sign matrices and on challenges related to Young tableaux. Applications in chemistry, synthetic biology and industrial applications. We will address problems in chemical graph theory and polyhedral self-assembly in synthetic biology. Knowledge will be applied in various business domains, such as mobility, fintech, etc.

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