Dušan Navrátil, the author of the dissertation has submitted a remarkable piece of work. Although it is apparent that the thesis is composed of a series of scientific papers connected by accompanying text, the resulting document is coherent, well-written, and presents interesting and original scientific findings. The central theme of the dissertation concerns Lie symmetries and their applications to ordinary and partial differential equations, with a particular emphasis on problems arising in continuum mechanics.

The first chapter serves as an introductory overview. In the second chapter, the theoretical foundations are reviewed: Lie groups, Lie algebras, the exponential map. The author introduces jet calculus, point transformations, the infinitesimal method, and group differential invariants. In the third chapter, the author demonstrates a strong command of jet calculus. This chapter is primarily technical in nature and includes a representation of the jet group of order three, which already provides sufficient insight into the general structure. The fourth chapter marks a skillful transition to applications. The discussion begins with an analysis of ordinary differential equations, starting with a third-order equation derived from a generalized lubrication model. The author investigates the symmetry properties of the equation y′′′=1/y^n for several notable values of n. These are valuable results obtained during a research stay with Professor M. C. Nucci. The next chapter is devoted to partial differential equations. First, the author addresses the classical and couple-stress theories of elasticity, followed by an analysis of Cosserat elasticity. The original results are expressed through a classification of infinitesimal symmetry generators, which can be regarded as valuable contributions to the theory of continua. The author then turns to the CHM equation, which is related to Rossby waves, and again provides a symmetry analysis by identifying the relevant generators. Continuum mechanics also serves as the conceptual basis for the final chapter, which introduces a helicoidal modeling approach. The concepts of oriento-position, roto-translation, and curvature are formulated geometrically using dual numbers (representing the simplest model of a Weil algebra). These results lead to a compact and geometrically consistent formulation of equilibrium equations for materials with microstructure (Cosserat media), and reveal a profound connection between symmetry, geometry, and the physical behavior of material systems.

The objectives and expectations of doctoral study have been fully met, and I consider the outcome to be highly successful. The dissertation is carefully prepared and virtually free of formal errors.

Some of Dušan Navrátil’s original papers have already been accepted for publication, and others are currently under peer review. In two cases, he is a co-author, but he wrote the majority of the papers himself, as the sole author. Throughout his doctoral studies, he has demonstrated a high degree of independence, originality, and attention to detail. His computing skills have also proven valuable, with some calculations requiring the use of specialized software. Overall, I am convinced that through this dissertation, Dušan Navrátil has not only demonstrated his ability to conduct scientific research, but has also taken a very promising step toward a successful academic career.

I wholeheartedly recommend the dissertation for defense.

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