# Research

I conduct research in mathematics and applied mathematics. My research is in the non-commutative geometry and analysis on the interface of algebra, geometry and analysis and their applications related to other topics.

**General discretisations of derivations (vector fields) with general dynamic control of discretisationsteps**

Here, we examine both the underlying algebraic structures for such discretisations and their applications and generalisations. Our work from 2003-2004 in this direction have been cited by other scientists worldwide as the work which laid the foundation for a new growing area of the algebra now called "Hom-algebra structures". The area is being developed by the researchers and their graduate students in France, USA, China, Luxembourg, Algeria, Tunisia and Norway. The main results, framework and new concepts and algebraic objects that we introduced are of fundamental importance and apparently of great interest outside of algebra, including the analysis of general dynamically controlled discretisations of differential equations and integral equations and related structures in topology and differential geometry, waveletanalys and harmonic analysis on fractals and algebraic methods for dynamic and iterative function systems, and for the development of new classes of adaptive methods in numerical analysis and computational methods, time series of complex stochastic systems, and development of new dynamic controlled discrete models in physics and technology. I therefore believe that research on "Hom-algebra structures" in analysis, dynamic systems, numerical methods and applications have great potential.

**Commuting operators in interaction with algebraic geometry and nonlinear equations**

Here, we investigate extensions of Burchnall-Chaundy theory about algebraic dependence of commuting differential operators to discretisations of differential operators and other operators, and commuting elements in the underlying algebraic structures. Further research in this direction may lead to the development of new methods for nonlinear differential and difference equations and the expansion of the interaction between non-linear equations and algebraic geometry of algebraic curves and varieties of other types of equations than differential equations.

**Commutative subalgebras in the non-commutative algebras and operator algebras**

Here we investigate commutative and maximal commutative subalgebras in non-commutative algebras, von Neumann algebras, C *-algebras, Banach and other normed algebras. Special focus we place on the interplay between commutative subalgebras, ideals and properties of dynamical systems and the actions of groups and semi-groups in non-commutative algebras of crossed product type which incorporate both the properties of space and properties by acting on the space dynamical systems, groups or semi-groups. Our results in this direction will contribute to research about several famous open problems such as the known Dixmiers problems in non-commutative algebra, Kadison-Singer problem in operator algebras, Feichtinger presumption of "frames" within waveletanalysis and further development of the deep interplay of operator algebras, dynamical system, wavelet analysis and fractal geometry.

**The theory of matrix monotone, matrix convex, operator monotone and operator convex functions (Löwner theory) and interpolation teory.**

Classes of matrix monotone and operator monotone functions (Pick functions) and matrix convex and operator convex functions form a natural extension of monotone and convex functions to functions of matrices and operators on Hilbert spaces with respect to the order relation on matrices and operators. Our results in this direction are about 30 year’s old problem to prove existence and explicitly construct functions that are in the gaps between these functional classes for matrices of different dimensions. We have shown that it is possible constructing infinitely many such functions and moreover polynomials in the gaps between classes with the help of solutions to so-called truncated moment problems. Explicit method we developed is based on in-depth analysis of Hankel matrices and explicit solutions of moment problems and thus leads to concrete ways to study how these classes of functions are different between each other and relate to other important function spaces. For example, we studied the interaction between these classes of functions and interpolation spaces. Interaction of matrix monotone, matrix convex, operator convex and operator monotone functions (Pick functions) and interpolation spaces from interpolation theory has important connections with several directions within the operator algebra and non-commutative geometry. Matrix monotone and Matrix convex functions have interesting applications in economics and finance (risk aversion), quantum computing and quantum information theory, and interpolation spaces are important for optimization, differential equations and non-commutative geometry. Further research on explicit constructions of matrix convex and matrix monotone functions based on the moment problems and interactions with interpolation can lead to the development of new methods in those areas of applications and to solutions of more open problems about inclusions of function spaces.

**I also do research on the applications of the above methods and results in
Information Technology and Computer Science**

Matrix Analysis, graph theory and Markov chains based methods and algorithms for internet and database search engines, and relevance ranking information (Page rank algorithms and other related algorithms ) and their applications in text mining and machine learning.

Quantum computers, quantum information and quantum computing (especially methods based on matrix analysis, operator theory and algebra)

**Medical informatics, bioinformatics and related fields**

Mathematical methods and algorithms for computer-based analysis of medical texts and medical terminology resources based on the combination of NLP (Natural language processing), graph theory and matrix analysis.

**Mathematical methods for financial and economic models**

Matrix analysis, Markov chains, differential and difference equations and stochastic modeling for prediction of behavior in financial markets, financial optimization, risk analysis and modeling of financial markets.

**Physics and related areas**

Development of mathematical models and algorithms for complex physical systems based on matrix analysis, automatic control and difference and differential equations for the analysis of complex physical systems.

Since my appointment as Professor in Mathematics/Applied Mathematics and the subject representative at Mälardalen University in mid-2011, I have also worked with

1) development of cooperation between Mathematics and the established strategic research environments at Mälardalen University;

2) development of research contacts and collaboration with colleagues in the School of Education, Culture and Communication and at other schools at Mälardalen University;

3) strengthening of the research base and research related to mathematical and other training programs at Mälardalen University by extending collaboration in research and education between mathematics and other subjects.