The main areas of current research conducted at the Faculty of Mathematics are grouped into several topics, which have been pursued for several years.
Department of Algebra and Geometry
In addition to research conducted at the Algebra Unit and the Geometry Laboratory, the research topics also cover non-commutative geometry, both differential and algebraic, and in particular algebraic structures, which play an important role in non-commutative geometry. In particular, symmetries in non-commutative geometry, quantum groups, intertwining structures, as well as geometric objects of special importance, such as non-commutative principal bundles or quantum homogeneous spaces, are studied. Issues of non-commutative singularity removal and smoothness of non-commutative geometric objects (e.g., orbifolds) are analyzed. Elements of non-commutative spectral geometry (à la Connes) are studied, in particular real structures on spectral triples. In addition, the research focuses on Hopf algebras, coproducts and their commodules, and ring-like algebraic objects such as braces and clamps.
Algebra Department
The research of the Algebra Department focuses on the properties of invariants of groups, commutative rings, and non-commutative algebras, mainly Hopf algebras. The scope of topics is broad and includes the following issues: ring extensions, radicals of rings and groups, filial rings, additive groups of rings, infinite Abelian groups, Abelian rings, non-distributive rings, subgroup lattices, automorphisms, and generators of finite groups.
Geometry Laboratory
The main research of the Geometry Laboratory focuses on projective geometry, its generalizations, and the geometry induced by quadrics in projective spaces. The spaces under consideration can be given the structure of a partial space of lines (bundle spaces - Grassmann spaces, higher-order quadrics, Segre products), or the structure of a Benz space (circle space). The main focus of the research is on characterization and classification issues, both of the more important classes of introduced spaces and of standard derivative constructs (subspaces, automorphisms, characteristic configurations, etc.). This focus also includes issues of immersion characterization - representation of spaces as immersed “manifolds.” One of the themes of this research is an attempt to find natural generalizations of classical concepts (e.g., parallelism, projectivity) and to identify their basic properties. Other areas of research include generalizations of hyperbolic geometry (quasi-hyperbolic geometry) and orthogonal geometry, as well as combinatorial methods for the construction of partial simple spaces.
Department of Analysis
Functional Analysis Laboratory
The research conducted by the Functional Analysis Laboratory is related to aspects of contemporary analysis that focus on the theory of nonlocal and singular operators and the equations they generate. The scope of topics is broad and includes the following issues: structures of operator algebras generated by the symbolic calculus of nonlocal pseudodifferential operators and boundary value problems, representations of operator algebras generated by automorphisms and endomorphisms, ergodic and entropy theory of nonlocal operators, topological and dynamic methods of calculating spectral characteristics, perturbation theory of nonlocal operators, theory of nonlocal equations in function spaces, differential operators with ?-potential and distribution multiplication problems, extensions of symmetric operators to self-adjoint ones. Methods and results of the theory of equations with nonlocal and singular operators can be applied in stochastic analysis, the theory of dynamical systems, the theory of pseudodifferential operators and convolution operators with oscillatory coefficients and operating in the area of complex equations with small parameters and resonances, thermodynamics, stochastic physics, and the theory of point interactions between molecules.
Differential Equations Laboratory
The scientific research conducted at the Differential Equations Laboratory is intertwined with the mainstream of global research on the qualitative and quantitative theory of ordinary differential equations, functional equations, and equations with random coefficients, commonly known as dynamic systems, which are widely used in modeling real-world processes. Among other things, we study the behavior of solutions of dynamical systems describing various random influences, including those related to periodic Markov processes, in order to obtain resonance conditions for solving problems in catastrophe theory, ecology, and oceanography.
We are also interested in studying dynamical systems on time scales. In particular, we seek to obtain necessary and sufficient conditions under which certain types of systems exhibit selected asymptotic properties of solutions. One of the themes of this research is the problem of consensus in multi-agent systems on any time scale.
The laboratory also conducts research on so-called C, S, and E functions, which are invariant under the action of Weyl groups associated with simple Lie groups and families of orthogonal polynomials generated by them, as well as research on the dependence of the weight function of orthogonal polynomials and the differential equation that defines a given family of polynomials. Such a relationship is known in the case of generalized Jacobi polynomials and generalized Legendre polynomials. This relationship is being studied in other, more general cases.
Department of Mathematical Physics
The scope of research conducted at the Department of Mathematical Physics covers broadly understood mathematical methods in physics. In particular, the applications of operator algebra theory and Banach differential geometry to the description of classical and quantum physical phenomena.
Department of Geometric Methods in Physics
The research conducted by the staff of the Department of Geometric Methods in Physics can be divided into the following topics:
Integral Systems Laboratory
Our research focuses on the relationship between bi-Hamiltonian structures and the integrability of Hamiltonian systems. These issues concern manifolds on which a pair of compatible Poisson brackets is defined (bi-Hamiltonian manifolds). For many classical integrable systems, two different Hamiltonians (or more) can be given that generate the same equations of motion by introducing different Poisson structures as a description of the problem. This allows us to determine the sequence of first integrals in involution, the Casimir function, and thus answer the question of the integrability of the system. In particular, the research focuses on issues related to the construction of multi-parameter families of Lie-Poisson systems through constructions related to contractions and deformations of Lie algebras.
Research is also conducted on the factorization method and its applications to functional equations, in particular differential equations that are discretizations of the Schrödinger equation.
In addition, research is being conducted on methods that allow for faster and more accurate processing of digital data (e.g., images, signals) sampled on fragments of 2-, 3-, and higher-dimensional networks on which Weyl groups (finite Coxeter groups) associated with simple Lie algebras operate. Research into the applications of non-crystallographic Coxeter groups in biology, particularly in virology.
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