Methods in nonlinear analysis, applications to differential and variational problems, and scientific computations

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In my research I am developing new mathematical methods to investigate the impact of symmetries on dynamical systems. One can easily observe that almost every design (e.g. buildings, networks, configurations of devices, etc.) has certain symmetries, which are expressions of our tendencies for elegance. Very little is known, how symmetries affect the performance or reliability of such design (e.g. the earthquake resistance of buildings, electric systems, chemical processes). Even in simple models there are difficult problems leading to complicated analysis involving advanced algebraic and analytic method. My research area, the Equivariant Analysis, is focused on creating new tools (for example the equivariant degree) allowing effective and complete study of symmetric dynamical processes. My interests are two folded: theoretical and applied. In the theoretical part, based on the most recent achievements in various mathematical fields (such as algebraic and equivariant topology, transformation groups, algebra, differential geometry, differential equations, functional analysis, etc.), I am trying to understand the complex nature of abstract nonlinear symmetric equations, so it becomes possible to extract essential information about the solutions and their symmetric properties. This information leads to a development of new computational methods (involving algebraic, topological and analytical routines). In the applied part, we establish standard settings and create comprehensive routines, which allow usage of computers (e.g. Maple package was created for several groups of symmetries) in order to provide the user with quick and precise way to analyze dynamical systems, making our theoretical achievements accessible to applied mathematicians and even engineers. Our methods were used, for example, to investigate symmetric configurations of (electrical, chemical or biological) oscillators, in population models, or electrical transmission lines.