February 28, 2023
Ara Balaki (University of Ostrava)
Introduction to cryptography
Abstract:
In this presentation we will first have a brief introduction to the history of cryptography with an emphasis of what is good cryptography and what makes a good cryptographic system, and then an overview of public-key cryptography with examples, and finishing with some notes about the importance of group theory in the study of such crypto-systems.
Slides
April 14, 2023
Ivan Kaygorodov (University of Beira Interior, Portugal)
Transposed Poisson algebras
Abstract:
Recently (2020), a dual notion of the Poisson algebra (transposed Poisson algebra), by exchanging the roles of the two binary operations in the Leibniz rule defining the Poisson algebra, has been introduced in the paper of Bai, Bai, Guo, and Wu. They have shown that the transposed Poisson algebra defined in this way not only shares common properties with the Poisson algebra, including the closure undertaking tensor products and the Koszul self-duality as an operad but also admits a rich class of identities. More significantly, a transposed Poisson algebra naturally arises from a Novikov-Poisson algebra by taking the commutator Lie algebra of the Novikov algebra. The talk is about new results related to transposed Poisson algebras and their relations with other types of algebraic systems.
Slides
April 24, 2023
Dmitry Gromov (University of Latvia)
Ranking nodes in signed networks: an algebraic perspective
Abstract:
This study is devoted to the problem of network ranking, which consists in determining the node hierarchy in a network. A number of algorithms have been proposed for the unsigned networks, i.e., networks whose edges have only positive weights. We mention the eigenvector centrality, the PageRank, and HITS, as well as their variants such as the weighted PageRank and others. These algorithms typically admit a neat algebraic formulation and can be studied using the methods from linear algebra and matrix analysis. We are particularly interested in the convergence properties of the algorithms, the existence of invariants, and the qualitative behavior of the solutions under the (structured) variation of parameters.
The situation becomes more involved if we wish to rank the nodes in a signed network, i.e., the network, whose edges can have both positive and negative weights taken from a finite set. The most common case is the binary set {-1,1}. Such networks describe not only the structure of interactions between different agents, but they also reflect the positive or negative relations between them. Because of this additional feature, signed networks are indispensable for analyzing social interactions in politically, religiously, racially or otherwise divided societies, the voting processes, and the competition/cooperation relations between companies, to mention just a few.
In the presentation, an overview of the previously obtained results will be given, and the novel results will be presented. In doing so, a main emphasis will be put on the algebraic structure of the considered methods.
Slides
April 25, 2023
Dmitry Gromov (University of Latvia)
Optimal control and value of information: theory and applications
Abstract:
Optimal control theory is an indispensable tool for the computation and analysis of optimal strategies in various applications, ranging from economics to technics. However, the optimal control methods can lead to wrong conclusions when applied to uncertain models. This particularly concerns the models that describe natural, social, or economical processes. In contrast to models built upon physical laws, the former models unavoidably incorporate uncertainties. These uncertainties result from the incomplete knowledge of the underlying dynamics of natural processes, the simplifying assumptions we have to make when describing complex phenomena, and our inability to measure the parameters of the considered models. Understanding and quantifying the model uncertainty is vital for making well-informed decisions.
Value of information (VoI) is a quantitative tool to analyze decision-making in the face of uncertainty. The first step of VoI analysis consists of formalizing the process of obtaining new information about the structure of the model, resp. values of parameters and quantifying the contribution of this new information toward obtaining better prediction of the model behavior. In the second stage, we carry out a qualitative and quantitative analysis of the profit acquired through the use of the new information. The estimation of the potential gain in profit is thus used to make an informed decision about the need for procuring more information.
In the presentation, a formal overview of different approaches aimed toward formalizing the value of information will be given. We will consider formal properties of the value of information and present several examples of the practical application of VoI.
Slides
December 13, 2023
Jan Gregorovič (University of Ostrava)
Invariants of curves in conformal manifolds
Abstract:
I will talk about invariants that can be assigned to curves in conformal manifolds of dimension greater than 2. An invariant is a quantity depending only on the curve and the conformal class of metrics and in particular, is invariant under all conformal transformation. The construction of these invariants uses the description of conformal manifolds via tractor bundles, which I describe in detail. Using tractor fields instead of vector fields along the curve allows to construct an analogy of the Frenet frame and use it to define invariants.
Slides
December 13, 2023
Kobiljon Abdurasulov (University of Beira Interior, Portugal)
The algebraic and geometric classification of nilpotent (binary, mono) Leibniz algebras
Abstract:
The complete algebraic and geometric classification of complex 5-dimensional nilpotent Leibniz and 5-dimensional nilpotent binary Leibniz and 4-dimensional nilpotent mono Leibniz algebras. As a corollary, we have the complete algebraic and geometric classification of complex 4-dimensional nilpotent algebras of nil-index 3.
Slides
