Symmetry, invariants, topology. Basic tools

Elementary concepts of group actions: orbits and their stabilizers, orbit types and their strata are introduced and illustrated by simple examples. We give the unified description of these notions which are often used in the different domains of physics under different names. We also explain some basic facts about rings of invariant functions and their module structure. This leads to a geometrical study of the orbit space and of the level surfaces of invariant functions (e.g. energy levels of Hamiltonians). Combining these tools with Morse theory we study the extrema of invariant functions. Some physical applications (not studied in other chapters) are sketched.