Scientific research

Scientific researches in the Institute of Mathematics are grouped into several main areas which correspond to researches developed in the Departments of the Institute. Clearly, due to a collaboration between people in the Departments, these areas may intersect in some points.

The scientific research in the Department of Algebra is related to group theory, ring and algebra theory (mainly associative). The main research interests concentrate around the following issues: finite groups, in particular the structure of finite groups, finite p-groups, automorphism groups, subgroup lattices of finite groups; radical theory of rings; filial rings; Hopf algebras, actions of Hopf algebras on associative algebras and their invariants.

Research in the Department of Applied Mathematics is focused on the study of connection between information theory and quantum game theory. Ideas of information theory are investigated in a wide range of classical and quantum models. Among others, we investigate transaction games in which well defined utility function enables to determine the most effective strategies in extreme conditions of deficit of information. Specific utility function (profit, profit intensity) based on the metrics on projective spaces which symmetries are natural among others for market models and the description of quantum systems. The discussed problems are of fundamental nature for both the information theory and description of capital markets. Solutions would permit to recognize the usefulness of measures of information for better understanding of quantum computing processes (quantum computer, quantum cryptography, modeling artificial intelligence in quantum formalism).

Scientific researches in the Department of Foundations of Geometry all concern projective geometry and its generalizations, with special emphasis on geometries induced by projective quadrics such as Grassmann spaces, quadratic Grassmann spaces, Veronese products, Segre products, spine spaces, and other partial linear spaces, and general Laguerre geometry.

Having so many different models at hand one can try to find generalizations, which may be of interest in foundations: weak theories of parallelism (affine partial linear spaces, quasi hyperbolic geometry), or weak theories of orthogonality for example. Other foundational question investigated in the Department concerns primitive notions which are sufficient to express considered geometries; in particular cases these are adjacencies, orthogonality (of subspaces), orthoadjacency, or plain incidence (on ``less classical" universes).

Some attention is also paid to "abstract" configurations on their own right, especially to combinatorial Grassmannians, combinatorial Veronesians, and multi-Veblen configurations.

The research of the Department of Functional Analysis is related to all the branches of modern analysis that are concentrated around the theory of non-local (especially weighted shifts) or singular operators and equations associated to them.

The following issues are within the wide scope of our interests:

• operator algebras associated to authomorphisms and endomorphisms,
• topological and dynamical methods of calculating spectral characteristics of non-local operators,
• ergodic theory and entropy,
• structure of operator algebras generated by the symbolic calculus of pseudo-differential and non-local operators,
• perturbation theory for non-local operators,
• non-local functional equations and differential operators with delta-potential,
• extensions of symmetric operators to self-adjoint ones.

The methods and results that we investigate and use in the theory of functional equations generated by non-local or singular operators have also application in stochastic analysis, dynamical systems, pseudo-differential operators and convolution operators with oscillating coefficients, and also in the theory of equations with small parameter and resonances, thermodynamics and stochastic physics.

The main directions of investigations carried out in the Department of Mathematical Analysis at the Chair is theory of ordinary and partial differential equations. The topics of the papers published by the members of the staff are as follows

• group-theoretical analysis of differential equations,
• generalized symmetries and constructions of solutions of nonlinear equations of Mathematical Physics,
• spectral and asymptotic analysis of differential operators,
• mathematical scattering theory of Schroedinger equation,
• hyperbolic systems of the partial differential equations and uniqueness and existence of the functional-differential equations,
• applications of the Legendre dual functional to the spectral exponent for some classes of the functional operators,
• harmonic analysis and its applications to the theory of special functions,
• stochastic analysis and its applications.

Researches in the Department of Mathematical Logic concern, primarily, mathematical logic in general and applications of logic in computer science and artificial intelligence. They are focused on the following, more particular topics: classical logic; algebraic logic; quantum logic; modal logics; computational complexity; formalization and automatization of mathematical reasoning; rough sets; approximate reasoning; knowledge discovery.

The main theme of researches developed in the Department of Mathematical Physics constitute algebraic and geometric methods in quantization, with particular interest in the class of C^*-algebras. Other subjects under development are:

• orthogonal polynomials in quantum optics,
• infinite dimensional integrable systems,
• Banach Lie-Poisson geometry, W*-algebras,
• quantization and coherent states,
• quantum logics, theory of measurement.

Some more problems are worth to mention, which were studied within the projects realized by the team:

• integrability of infinite Toda lattice by the construction of action-angle variables,
• relation of integrability of multiboson systems with the theory of orthogonal polynomials,
• infinite dimensional integrable systems on the restricted Grassmannian and their quantization by means of coherent state map,
• coherent state map and logics related to W*-algebras,
• quantum complex Minkowski space and other quantum phase spaces.

In the Department of Topology and Differential Geometry investigated are geometrical structures defined on complex manifolds and on holomorphic vector bundles by Bergman sections which includes researches of Kähler, Riemann and symplectic structures defined on complex manifolds, hermitian structures, curvatures and connections defined on holomorphic vector bundles and mappings of manifolds and bundles in universal objects such as Hilbert and projective spaces and connected with them natural bundles. Also investigated are grupoid differential structure of grupoid charts, bundles associated with differential grupoids, examples of differential grupoids.