The concept of transformation group is a mathematical formalization of the general philosophical idea of symmetry of the world. One may say that the world is cognizable inasmuch as it is symmetric. This principle is vividly reflected in the famous theorem of Emmy Noether establishing connection between symmetries of a physical system and conservation laws for physical quantities.
In mathematics, the theory of transformation groups is a broad area covering different branches of algebra, geometry and topology, dynamical systems, and mathematical physics. Essentially it gives rise to the origins of group theory and to the Erlangen program of Felix Klein, who proposed to consider various geometries from the viewpoint of their symmetry groups. Although transformation groups do not fit in a unified mathematical theory in a strict sense and different fields in this area may be rather far from each other, there are general consolidating principles and concepts, which make interosculation and mutual enrichment with ideas possible. This makes interaction and collaboration of researches in various fields related to transformation groups relevant and fruitful.
The Moscow school of transformation groups was established at Moscow State University by Ernest Vinberg (1937-2020) and Arkadiy Onishchik (1933-2019).