Team: Josef Šlapal (leader), Michael Lieberman, Jan Pavlík

The research group deals with the study of algebraic and topological methods with with respect to their applications in various areas of mathematics (homotopy theory, mathematical logic - especially model theory, discrete mathematics, etc.) and computer science (program verification, digital image processing, etc.). The specific research interest of individual members of the group is as follows:

In a series of articles focused on digital topology, which is an important branch of discrete geometry, J. Šlapal proposed new structuring of the digital plane and digital space based on the use of closure operators (more general than the Kuratowski closure operators), which provide a suitable notion of connectedness for the study and processing of digital images . While he has so far dealt with 2D images, his further research will be aimed at finding tools suitable for studying and processing 3D digital images. He will also continue his research in categorical topology, where he will study topological structures on categories such as closed spaces, convergence structures, neighborhood structures, quasi-uniform structures, etc. Among other things, he will focus on the Cartesian closedness of categories with topological structures (Cartesian closed categories have significant application in modern informatics). He is currently engaged in the study of topogenous orders on categories.

J. Pavlík achieved valuable results in the study of generalized metric spaces from the point of view of their use in designing new methods of processing digital images. These are methods based on thresholding using generalized ultrametrics, which make it possible to characterize objects of various properties in digital images. J. Pavlík further develops these methods, from a general point of view, and studies their mathematical background. Closely related to this study are unsolved problems on the border between the theory of graphs and that of metric spaces, the solution of which will also be focused on (e.g. a generalized approach to graphs and metric spaces, representation of graphs using generalized metrics, distributivity of segmentation spaces, topography of generalized metric spaces, etc.).

With M. Lieberman, who has achieved valuable results in the study of algebraic methods with a focus on applications in logic, the group is connected mainly by using the tools of category theory in solving algebraic and topological problems. In recent years, he managed (in collaboration with J. Rosicky and S. Vasey) to generalize the model-theoretic concept of stable independence (including, for example, algebraic and linear independence) to the context of abstract categories. This generalization showed, among other things, the close relationship between dependence and abstract homotopy theory. It also led to a better understanding of the fine structures of module categories, and in his ongoing research, M. Lieberman will focus on applications of the obtained results in graph theory. It turns out that apparently simple categorical arguments are sufficient to determine the limits of the complexity of axiomatizing objects that appear in mathematical analysis and physics (e.g. Banach spaces), thus clarifying the logical tools suitable for their study. M. Lieberman will also deal with this topic.

The affinity of the research orientations of all three members of the group resulted in the preparation and submission of a joint project to the GA CR in 2021 entitled "Categorical factorization systems and their applications in algebra and topology", which, however, was not accepted for funding. M. Lieberman therefore submitted a new project to the GA CR in 2024 (with a foreign collaborator). We will strive to obtain other scientific projects, including foreign ones. The research group will continue to deepen and expand cooperation between its members and cooperation with similarly focused experts from Czech and foreign universities. It will also be temporarily staffed by experts from abroad who will stay at the Institute of Mathematics as part of various projects. We will continue to publish the achieved results in high-quality scientific journals and at prestigious international conferences. We will focus on listing thesis topics for the Mathematical Engineering degree program as well as doctoral study topics in the Applied Mathematics program in order to recruit students for group work. We will also participate in the promotion of the Mathematical Engineering study program with the aim of obtaining as many quality students as possible for this program.