ABSTRACT
This chapter is devoted to the proof of René Thorn’s theorem classifying the degenerate critical points of functions whose codimension is at most 4. Roughly speaking, this theorem states that there are only seven different types of degenerate critical points for such functions; these are often referred to as the 7 elementary catastrophes. Using similar arguments, this theorem can be generalized to cover the critical points of functions with codimension at most 6; see Exercises 6.2 and 6.3. In this case, the classification gives 14 different standard types of such critical points. Furthermore, for functions of just two variables whose codimension is at most 7, a classification can also be achieved resulting in 17 different possibilities; see Exercise 6.4. A finite classification is no longer possible, however, for functions of two variables whose codimension exceeds 7 or for functions of more than two variables with codimension at least 7, as shown in Exercises 6.6 and 6.7.
