Mathematics (Md)
Differential invariants
SK
The thesis is based on the theory of differential invariants contained in the publication [1]. A focus of the thesis are differential invariants of tensors. The thesis also deals with the basic consequences of invariance in global variational geometry, such as conservation laws for extremal equations. The goal is to find assertions on the structure of all differential invariants of the metric and the given tensor field. The obtained results can be used for finding invariant Lagrangians for different types of physical fields.
[1] D. Krupka, J. Janyška, Lectures on Differential Invariants, Folia Facultatis Scientiarum Naturalium Universitatis Purkynienae Brunensis, Mathematica 1, University J. E. Purkyně, Brno, 1990. [2] D. Krupka, Introduction to Global Variational Geometry, Atlantis Studies in Variational Geometry, D. Krupka, H. Sun (Eds.), Atlantis Press, 2015. [3] J. Brajerčík, Second order differential invariants of linear frames, Balkan J. Geom. Appl., Vol. 15, No. 2 (2010) 22-33 (electronic version)
doc. Mgr. Ján Brajerčík, PhD.
Mathematics (Md)
Classification of aggregation functions on ordered sets
SK
The objective of this work is the axiomatic study and analysis of aggregation functions defined on ordered sets. The aim is to classify these functions with regard to the algebraic properties of the structures on which they are defined and their possible applications in decision theory.
[1] Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions. Cambridge University Press, Cambridge (2009). [2] Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners. Studies in Fuzziness and Soft Computing, vol. 221. Springer, Heidelbeg, (2007). [3] Halaš, R., Pócs, J.: On the clone of aggregation functions on bounded lattices. Inf. Sci. 329, 381–389 (2016).
RNDr. Jozef Pócs, PhD.
Mathematics (Md)
Combinatorial properties of real line
SK
The aim of the work is to study relational systems on the set of real numbers and their cardinal characteristics using methods of forcing, models of set theory, and descriptive set theory.
[1] Bartoszynski T., Judah H., Set Theory. On the Structure of the Real Line, A.K. Peters, Wellesley, MA, 1995. [2] Bukovský L., The Structure of the Real Line, Monografie Matematyczne, 71. Birkhäuser/Springer Basel AG, Basel, 2011. [3] Jech, T., Set Theory, The Third Millennium Edition, revised and expanded, Springer, 2002.
doc. RNDr. Miroslav Repický, CSc.
Theory and Practice in Mathematics Teaching (TVMd)
Covariational reasoning in learning functions
SK
Research on the development of students’ functional thinking highlights the importance of viewing a function graph as a representation of how the values of two quantities change simultaneously. For this reason, the present study focuses on identifying the ways of thinking that Slovak students use when imagining and representing the simultaneous change of two quantities, as well as the ways of thinking that enable or, conversely, hinder them in constructing and interpreting graphs as representations of such simultaneous change. The study also examines how Slovak students apply their covariational reasoning when working with different representations of a function (table, formula, graph) and across various contexts. In doing so, it aims to contribute new insights into the development of covariational reasoning and to provide implications for the teaching of functions with an emphasis on the covariational aspect.
Based on relevant literature and current research findings, to identify the ways of reasoning that Slovak students use when conceptualising and representing the simultaneous change of two quantities. To identify the factors and modes of reasoning that either support or hinder students in constructing and interpreting graphs as representations of the simultaneous change of two quantities. To investigate how Slovak students apply covariational reasoning when working with different representations of a function (graphical, tabular, algebraic, and verbal). To design and implement an instructional experiment aimed at developing covariational reasoning as a form of conceptual understanding of functions.
doc. RNDr. Ingrid Semanišinová, PhD.
Mathematics (Md)
Qualitative properties of nonlinear functional differential equation
SK
The objective of this thesis is to study qualitative properties (asymptotic behavior, oscillation) of functional differential equations. The aim is to establish sharp criteria for broad classes of nonlinear equations with p-Laplacian-type operators
[1] Berezansky, Leonid, Alexander Domoshnitsky, and Roman Koplatadze. Oscillation, nonoscillation, stability and asymptotic properties for second and higher order functional differential equations. Chapman and Hall/CRC, 2020. [2] Došlý, Ondřej, and Pavel Řehák. Half-linear differential equations. Vol. 1000. Elsevier, 2005. [3] Jadlovská, Irena, and Agacik Zafer. "Oscillation Theorems for Second‐Order Trinomial Delay‐Advance Differential Equations." Mathematical Methods in the Applied Sciences, 2026, in press
Ing. Irena Jadlovská, PhD.
Mathematics (Md)
Mathemematical theory of voting and elections
SK
Investigate the computational complexity of the problem of computing the possible and necessary winner for various voting systems, given different types of incomplete information. Design efficient algorithms for special cases of input data.
1. Boutilier, C., Lang, J., Oren, J., & Palacios, H. (2014). Robust winners and winner determination policies under candidate uncertainty. In Proceedings of AAAI-14. 2. Cechlárová, K., Lesca, J., Trellová, D., Hančová, M., & Hanč, J. (2023). Hardness of candidate nomination. Autonomous Agents and Multi-agent Systems, 37(2), 1–33. 3. Chakraborty, V., Delemazure, T., Kimelfeld, B., Kolaitis, P. G., Relia, K., & Stoyanovich, J. (2021). Algorithmic techniques for necessary and possible winners. ACM/IMS Transactions on Data Science, 23, 1–23. https:// doi. org/ 10. 1145/ 34584 72 4. Faliszewski, P., Gourvès, L., Lang, J., Lesca, J., & Monnot, J. (2016). How hard is it for a party to nominate an election winner? In Proceedings of IJCAI-16 (pp. 257–263).
prof. RNDr. Katarína Cechlárová, DrSc.
Mathematics (Md)
Mathematical foundations of hierarchical data aggregation
SK
The thesis is focused on the theoretical research of nonlinear aggregation operators within hierarchical structures. The primary objective is the systematic study of the mathematical properties of iterated integrals and related aggregation mechanisms in multi-level data aggregation. In complex decision-making settings, criteria are often organized into multiple hierarchical levels (e.g., technical parameters – operational efficiency – overall value). The work will particularly focus on the analysis of conditions under which interactions (synergy and redundancy) among criteria can be consistently propagated from lower to higher levels, as well as on the study of the properties of the resulting hierarchical models. Attention will be paid to issues of stability, robustness, and representation of these models, together with the identification of conditions suitable for their reliable use in decision-making tasks.
prof. RNDr. Ondrej Hutník, PhD.
Mathematics (Md)
Nonlinear Dynamics and Predictive Inference under Non-additive Measure Frameworks
EN
The dissertation examines dynamical systems driven by non additive measures, addressing limitations of classical theories that rely on additivity and therefore fail to capture interaction driven behaviour in complex phenomena. Whereas traditional models often treat nonlinear interactions as anomalies, this work proposes a conceptual shift toward a unified, statistically rigorous framework for non additive, interaction aware system evolution. The core contribution lies in establishing a coherent mathematical and inferential framework in which non additive, interaction driven dynamical systems can be systematically analysed. This represents a qualitative transition in the field, providing the theoretical foundations necessary to study systems whose components do not merely accumulate but interact in inherently nonlinear ways. The research builds on mature developments in aggregation operators, Choquet type models, and aggregation based representations of uncertainty. By extending established constructions from uncertainty modelling and nonlinear inference, the thesis develops new inferential tools and analytical techniques. The outcome should be a unified theoretical framework that advances isolated methodological results toward a deeper understanding of nonlinear, interaction driven dynamical phenomena, with potential applications across mathematics and related scientific domains.
1. B. Piccoli. Measure differential equations. Arch Rational Mech Anal, 233:1289–1317, 2019. 2. Negi, S. S., and Torra, V. A note on Sugeno exponential function with respect to distortion. Applied Mathematics and Computation 470 (2024), 128586. 3. Ontkovičová, Z., and Torra, V. Computation of Choquet integrals: Analytical approach for continuous functions. Information Sciences 679 (2024), 121105. 4. J. Kurzweil. Generalized Ordinary Differential Equations: Not Absolutely Continuous Solutions. Series in Real Analysis. World Scientific, 1 edition, 2012.
doc. Mgr. Jozef Kiseľák, PhD.
Mathematics (Md)
Schwarzschild metric and its generalization
SK
The thesis deals with the mathematical foundations of the general relativity. The starting point is the structure of the Schwarzschild metric and the underlying geometric structures of the general relativity. The goal is to obtain assertions on the generalization of the Schwarzschild metric to metric dependent on velocities (Finsler metric) invariant to the action of the Lie groups of general relativity.
[1] De Felice, F.; Clarke, C.J.S. Relativity on Curved Manifolds. In Cambridge Monographs on Mathematical Physics; Cambridge University Press: Cambridge, UK, 1990. [2] D. Krupka, Introduction to Global Variational Geometry, Atlantis Studies in Variational Geometry, D. Krupka, H. Sun (Eds.), Atlantis Press, 2015. [3] D. Krupka, J. Brajerčík, Schwarzschild Spacetimes: Topology. Axioms 2022, 11 (12) 693. https://doi.org/10.3390/axioms11120693
doc. Mgr. Ján Brajerčík, PhD.
Mathematics (Md)
Testing of multivariate random variables with special variance structures
SK
Study the tests of the mean and variance parameters in models with elliptical distributions and with special variance structures, their derivation and properties.
1. Gupta, A.K., Nagar, D.K. (1999): Matrix variate distributions, Chapman and Hall 2. Gupta, A.K., Varga, T., Bodnar, T. (2013): Elliptically Contoured Models in Statistics and Portfolio Theory, Springer 3. Fang, K.-T. and Zhang, Y.-T. (1990). Generalized multivariate analysis. Springer-Verlag, Berlin 4. Filipiak, K., Klein, D., Mazur, S., Mrowińska, M. (2025). Likelihood ratio test for covariance matrix under multivariate t distribution with uncorrelated observations, Journal of Multivariate Analysis 210, no. 105490. 5. Yao, J., Zheng, S., Bai, Z. (2015). Large Sample Covariance Matrices and High-Dimensional Data Analysis. Cambridge University Press.
doc. RNDr. Daniel Klein, PhD.
Mathematics (Md)
Distance-constrained realizations of geometric graphs
SK
The aim of the project is to study graph representations in Euclidean spaces (or their substructures), whose edges are formed by segments, while their lengths can only take values from a prescribed set of admissible lengths. This concept includes, for example, unit-distance graphs in the plane (studied, for example, in connection with Nelson's problem of the chromatic number of the Euclidean plane), odd graph drawings or integer/rational drawings of plane graphs; in the case of the requirement of equality of the lengths of all edges of the graph, the minimum dimension of the Euclidean space in which the graph can be realized in such a way was also investigated (the so-called graph dimension; a related concept, where vertices not forming an edge should have a distance different from the length of the edges leads to the definition of a similar invariant, the so-called Euclidean dimension of the graph). Attention will be focused on obtaining new knowledge on unit, odd or prime geometric drawings of graphs of specific classes (convex polyhedra, planar graphs, graph operations, or chromatically constrained graphs) in the plane or in higher-dimensional spaces, as well as research of properties of analogies of the graph dimension taking into account non-trivial symmetries of multidimensional geometric graph representation.
[1] Alexander Soifer: The Mathematical Coloring Book, Springer, 2009 [2] Alexander Soifer: The New Mathematical Coloring Book, Springer, 2024 [3] H. Ardal, J. Maňuch, M. Rosenfreld, S. Shelah, J. Stacho: The Odd-Distance Plane Graph, Discrete Comp. Geom. 42 (2009) 132-141
prof. RNDr. Tomáš Madaras, PhD.