General interests

• Non-linear Dynamical Systems. Non-linear partial differential equations are central to mathematics, pure and applied. They provide an universal approach to describing almost any physical system which depends on continuously varying independent variables. Their relevance is growing, powered by modern computational equipment, capable of solving suitably discretized approximations of the equations. Equally important is the impact the non-linear partial differential equations have on fundamental mathematics, opening new branches, cross-fertilising existing ones, benefiting from the results. In addition to numerical methods, I am also interested in algebraic, topological, and group methods in the theory of dynamical systems.

• Applications of Differential Equations to Physical Systems. Perturbation methods; Instability of fluid flow and in magnetohydrodynamics; Pattern formation out of statistical equilibrium; Rotating thermal convection; Dynamo theory; Geophysical fluid dynamics; Cardiac modelling.

• Recreational & Technical. Algorithms; Programming techniques, languages and tools; Computer Algebra Systems; Visualization.

The pursuit of my research interests would not be (have been) possible without the support of the following foundations and universities:

Work

• Excitation and propagation of waves in cardiac tissue and other excitable media. Through coordinated contraction of the heart, blood is pumped to all parts of the human body ensuring our health and vitality. This mechanical activity of the heart is controlled by electrical impulses, called action potentials, which propagate along the membranes of the cardiac cells. An irregular action potential propagation leads to irregular mechanical activity of the heart. Such conditions are known as cardiac arrhythmias. Cardiac arrhythmias are abnormal often life-threatening events. Some of them can lead to sudden cardiac death which is the leading cause of death in the industrialised world. Mathematically, the normal as well as the abnormal propagation of cardiac action potentials can be modelled by a system of partial differential equations. Thise equations are proposed on the basis of physiological measurements and differ for the various types of cardiac tissues. The contemporary cardiac models are remarkably detailed and provide impressive accuracy. They are very complicated and consist of a large number of equation and thus by necessity they have to be solved numerically. However, simplifications of the detailed cardiac models exist. Such simplifications can be achieved by taking into account the asymptotic structure of the detailed models. For additional details on the subject in general and on my work in particular see my list of publications.

• Convection and magnetic field generation in rotating spherical fluid shells. The Earth has a substantial magnetic field. The existence of this field was one of the first global properties of the planet to become known to man. The geomagnetic field has been of much historical importance because of the role of the magnetic compass in exploration of the planet. The Earth's magnetic field has fascinated scientists for centuries and has, in fact, been the subject of one of the first truly modern scientific studies (W Gilbert,``De Magnete'', 1600), predating even Galileo and Newton. The main magnetic field of the Earth, as well as those of a great variety of cosmical objects, are generated by motions of conducting fluid in the outher core of the Earth. The motion of the fluid is due to convection powered by chemical processes, radioactivity and gravitational motions of the planet. These processes are modeled by a combination of the equations of fluid dynamics, thermodynamics, and electromagnetism. The models vary in complexity but for a realistic description they need to be three-dimensional and nonlinear. This means that they need to be solved mostly by numerical methods. Efficient numerical codes have been developed in the last 15 years which are capable of simulating the Earth's main magnetic field. The simulations however are expensive and time consuming which means that many open questions remain to be studied. My research on the topic involves using existing numerical codes for large scale computations, often in parallel environment, to study various processes of geomagnetic field structures and their genration and decay as function of the basic parameters of the problem. I am interested in modelling various secondary effects such as anisotropy of Earth's material properties, different heating models, compositional convection and others. Of special interest are the possibilities of comparison between numerical simulations and various geomagnetic observational phenomena such as field polarity reversals, secular variation impulses and others. For additional details on the subject in general and on my work in particular see my list of publications.