Ongoing projects
Quantum geometric theory of representations and non-commutative fiber bundles
The aim of the project is to study the properties of a special class of fibrations—whose fibers have rich symmetry—and thus attempt to solve important problems in representation theory, topology, and quantum group geometry.
Completed projects
Banach Poisson-Lie groups and integrable systems
The grant concerns the study of structures appearing in infinite-dimensional geometry. The focus will be on Poisson structures and related integrable systems. A model example of a manifold relevant to this topic is the Sato grassmannian. It is a Hilbert manifold additionally equipped with a natural Kähler structure. It is related, among others, to integrable systems such as KdV. The aim of the grant is to investigate in greater depth the geometric structures on infinite-dimensional manifolds modeled on both Banach and Frechet spaces.
Study of the structure of ties and their applications in ring and bracket theory
The aim of the project is to comprehensively study the properties of new algebraic structures based on sets equipped with a three-argument operation combining addition and subtraction, and multiplication separate from this operation. These structures, first defined by the project leader in 2017, combine the features of rings and currently intensively studied brackets, based on two associated group operations, and are called binders. The research undertaken by the team as part of the project includes both a detailed analysis of the abstract properties of ties and the construction of specific examples, their classification, and applications in the theory of rings, groups, and brackets.
Excellent Science – Support for Scientific Conferences
As part of the Ministry of Education and Science's “Excellent Science - Support for Scientific Conferences” competition, the Faculty of Mathematics received funding in the amount of PLN 101,970. These funds will be used to organize the 40th anniversary “Workshop on Geometric Methods in Physics” conference. The aim of the project is to subsidize the organization of a scientific conference presenting scientific achievements, including the latest results of scientific research or development work.
Invariant dynamics and Cartan C*-algebras: their generalizations, properties, and applications
Dr. Bartosz Kosma Kwaśniewski received funding in the amount of PLN 221,580 as part of the OPUS 18 competition announced by the National Science Centre for the implementation of the project “Non-commutative dynamics and Cartan C*-algebras: their generalizations, properties, and applications.” The research topic lies at the intersection of three rapidly developing branches of contemporary mathematics: operator algebra theory, dynamical systems, and operator theory. The project plans include cooperation with scientists from Germany, Brazil, Belarus, and Poland, the promotion of a doctor of mathematical sciences, and a series of scientific articles.
For scientists from Ukraine to continue their research in Poland
Grant under the National Science Centre program for researchers from Ukraine to continue their research in Poland. The aim of the program is to provide financial support to researchers by creating opportunities for them to be employed for one year and continue their scientific work in Poland. The Faculty of Mathematics' cooperation with Ms. Irada Dzhalladova consists of active participation in scientific seminars of the Differential Equations Laboratory, faculty seminars, as well as a series of lectures on topics such as: Information Security Economics and Econophysics.
Symmetry methods for differential equations and their discretization
New families of special functions of several variables and their properties
The immediate goal of the project is to create a Polish-Czech research team, which will bring measurable benefits to both parties participating in the project. The team thus created will study new families of special functions, related orthogonal polynomials, and their properties. The research will focus on open problems in this field, namely generating functions and orthogonality (i.e., perpendicularity) associated with families of orthogonal polynomials, rules for transforming families of special functions and families of polynomials between each other, comparison of the effectiveness of approximations using special functions, reduction of these families and corresponding polynomials. Each of the results achieved will be considered in terms of its applications in digital data processing, in particular for image (photo) analysis in cameras and mobile phones. In recent times, with digitization permeating almost all areas of human activity, the need for digital data processing is an extremely topical issue. The study of trigonometric and exponential functions and their generalizations, which enable the construction of discrete (stepwise) Fourier analysis, provides new mathematical tools for faster and more accurate image processing.
Analysis of constructions and structures related to the study of C*-dynamic systems
The project is related to the problem of developing a mathematical model of complex physical processes, including quantum processes, containing dynamic components and interactions with the external environment. In a mathematical model corresponding to such processes, two factors must be taken into account: the laws of motion (deterministic or stochastic) and the influence of the external environment. This means that motion should be described by an appropriate dynamic system or stochastic process, and interactions between molecules and the external medium are given by a certain potential. In addition to their application aspect, the objects studied in this project play a fundamental role in invariant harmonic analysis, mathematical physics, and specifically in quantum physics. In the Hilbert case, the operator algebras (C*-algebras) generated by the operators whose properties we are studying are called algebras associated with dynamical systems, covariance algebras, or cross products, and constitute one of the most important classes of C*-algebras defined by relations. Among the classical objects of this type, the best known are Cuntz algebras, or Cuntz-Krieger algebras, which, in addition to their dynamic connotations with topological Markov chains, have a natural interpretation in the language of graph theory, and currently generalizations of Cuntz-Krieger algebras are usually called C*-algebras associated with graphs.
Structures of C*-algebras defined by relations, spectral and ergodic properties of operators generating dynamical systems
The aim of the project is to develop new methods for studying, constructing, and describing the structure of C*-algebras defined by broadly understood relations (including relations of a dynamic, combinatorial, or stochastic nature) and, on this basis, to expand the spectral theory of a broad class of operators, in particular functional operators and weighted composition operators. The entire scope of the project concerns basic research: it is intended to make a significant contribution to the general theory of C*-algebras, in particular to the theory of universal C*-algebras, and to the contemporary spectral theory of functional operators. The project is related to the problem of developing a mathematical model of complex physical processes, including quantum processes, containing dynamic components and interactions with the external environment. Examples include particle motion and transformation processes. These particles can be neutrons, molecules, individuals in a biological population (e.g., infected individuals in epidemiological models), components of a chain reaction, etc. In a mathematical model corresponding to the above processes, two factors must be taken into account: the laws of motion (deterministic or stochastic) and the influence of the external environment. The objects studied in the project make it possible to take both of these factors into account, and the asymptotic and ergodic properties of the systems under consideration are described by the spectral properties of the corresponding operators. The description of such properties, which is one of the objectives of the project, is a fundamental problem of the qualitative description of a given process and is closely related to technological issues.
Banach Lie-Poisson spaces, integrable systems, and quantization
Operator algebras generated by dynamical systems: spectral, asymptotic, and entropic characteristics
Noncommutative Kahler structures
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