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

在我的研究中，我正在开发新的数学方法来研究对称性对动力系统的影响。人们可以很容易地观察到几乎每一种设计(例如建筑、网络、设备配置等)具有一定的对称性，这是我们追求优雅的倾向的表现。人们对对称性如何影响此类设计的性能或可靠性(例如，建筑物、电力系统、化工过程的抗震)知之甚少。即使在简单的模型中，也有一些困难的问题导致复杂的分析，涉及到先进的代数和分析方法。我的研究领域，等变分析，专注于创造新的工具(例如等变度)，允许有效和完整地研究对称动力过程。我的兴趣有两层：理论和应用。在理论部分，基于各种数学领域(如代数和等变拓扑、变换群、代数、微分几何、微分方程、泛函分析等)的最新成果，我试图理解抽象的非线性对称方程的复杂性质，从而能够提取关于解及其对称性质的基本信息。这些信息导致了新的计算方法的发展(包括代数、拓扑和分析例程)。在应用部分，我们建立标准设置和创建综合例程，允许使用计算机(例如，Maple Package是为几组对称创建的)，以便为用户提供快速而精确的方法来分析动力系统，使我们的理论成果为应用数学家甚至工程师所用。例如，我们的方法被用来研究(电的、化学的或生物的)振荡器的对称配置，在人口模型中，或在电力传输线中。

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.