Mathematics

The subject of our research is the application of methods of modern functional analysis, the theory of operator algebras and linear algebra to the description of physical phenomena. In particular, we study

• the notion of quantum entanglement,
• entanglement witnesses,
• quantum correlations,
• steering inequalities.

Our goal is also to describe the structure of positive mappings on matrix algebras. In particular we are focused on the problem of NPT bound entanglement.

Integrability of Distributions
Random Matrix Models. Orthogonal Polynomials
Differential Equations Through Symmetry
Control Theory

Porous medium equation.
Navier-Stokes equations.
Elliptical and non-linear parabolic equations.
Partial differential equations in Orlicz-Sobolev spaces.
Primitive equations of the ocean.
Stochastic risk assessment of nuclear waste storage.
Mathematical tools for the extraction, manipulation and prediction of information in databases.
Turbulence modelling.
Thermistor problem.
Formal processing of relational information systems.
Heat treatment of steel.
Fuzzy sets. Fuzzy logic.

- Numerical linear algebra (accurate and fast matrix algorithms, solving eigenvalue and singular value problems, perturbation theory, eigenvalue and singular value problems for matrices with elements from noncommutative fields)

APPLICATIONS OF ALGEBRO-GEOMETRIC TECHNIQUES: K-THEORY  IN REGULAR RINGS, C * -ALGEBRAS AND  LEAVITT PATH ALGEBRAS, THEORY OF LIE; ALGEBRAIC GEOMETRY, HOLOMORPHIC FOLIATIONS AND MODULAR CURVES.
FACTORIZATIONS IN NUMERICAL SEMIGROUPS AND AFFINE SEMIGROUPS.

Series and summability in normed spaces
Topological modules
Composition Operators
Lipschitz Spaces
Bishop-Phelps-Bollobás Property
Measure Theory in effect algebras
History of Mathematical Analysis

Group transformations:symmetries and equivalence transformations.
Conservation laws.
Nonclassical method. Potential symmetries.
Mathematical modelling of chemotherapy resistance in tumor development.
Exact solutions,Quañitative analysis.
Nonlinear waves and solitons.
Nonlocal symmetries and nonlocal related systems.
Diffusion systems: bacterial growth, tumor evolution. Other models from physics, medicine, engineering and economy.

Probability distributions, stochastic orderings, survival analysis,  exploratory data analysis, mathematical programming, stochastic models in finance and insurance, stochastic comparisons of actuarial and financial risks.

-          new estimates of Shannon's entropy obtained from the known properties of superquadratic functions,

-          new estimates of the relative entropy derived from Jensen's inequality for superquadratic functions,

Central focus of the research of this group includes graph theory (with emphasis on extremal graph theory), complex networks (with emphasis on multiplexes, curriculum networks and network descriptors), mathematical chemistry (with emphasis on molecular descriptors, nanotechnology, peptide antibiotics), and discrete mathematics in general. Our goal is to obtain purely mathematical results and multidisciplinary applications.