Consider a smooth (C ∞) function f: R n → R m and assume that f has a critical point at the origin, i.e., df (0) = 0. The theory of singularities as developed by Thom, Mather, Arnol’d, and others (Lu, 1976; Gibson, 1979; Arnol’d, 1981) addresses the following basic questions:

    (1) What is the local character of f in a neighborhood of the critical point? Basically, this question amounts to asking “at what point is it safe to truncate the Taylor series for f?” This is the determinacy problem.
     
    (2)  What are the “essential” perturbations of f? That is, what perturbations of f can occur that change the qualitative nature of f and that cannot be transformed away by a change of coordinates? This is the unfolding problem.
     
    (3) Can we classify the types of singularities that f can have up to diffeomorphism? This is the classification problem.