Catastrophe Theory Concepts

• The Classification of Elementary Catastrophes of Codimension ≤ 5 were written by Christopher Zeeman, from the University of Warwick, in Coventry, England.
• These notes were written and revised by David Trotman.
• The purpose of the notes is to give a minimal complete proof of the classification theorem from first principles.
• All results are proved, including those which are not standard theorems of differential topology.
• The theorem is stated in Chapter 1 in a form useful for applications.

Elementary Catastrophes

• Elementary catastrophes are singularities of smooth maps R^r -> R.
• They arise from stationary values of r-dimensional families of functions on a manifold.
• They also come from fixed points of r-dimensional families of gradient dynamical systems on a manifold.
• They are central to the bifurcation theory of ordinary differential equations.
• The case where r = 4 is important for applications parametrized by space-time.
• René Thom realized elementary catastrophes and recognized their importance.
• By 1963, Thom realized they could be finitely classified for r ≤ 4 with polynomial germs (x^3, x^4, x^5, x^6, x^3 + xy^2, x^2 + y^4).
• Thom sources of inspiration included: Whitney's paper on stable singularities for r ≤ 2, his work extending these results to r > 2, light caustics, and biological morphogenesis.
• Thom conjectured the classification, but it took years to prove as mathematics needed to develop necessary tools.
• Catastrophe theory stimulated developments in bifurcation, singularities, unfoldings, and stratifications.
• The heart of the proof is the concept of unfoldings, due to Thom.
• The key result states that two transversal unfoldings are isomorphic, necessitating a C∞ version of the Weierstrass preparation theorem.
• Malgrange proved this around 1965, with simpler proofs contributed by Mather and others.
• The proof in Chapter 5 is mainly from [1].
• The preparation theorem synthesizes analysis into an algebraic tool, and enables construction of the geometric diffeomorphism required to prove two unfoldings equivalent.
• John Mather first wrote explicit construction and proof of the classification theorem around 1967.
• The essence of the proof is contained in his published papers about more general singularities.
• In 1967, Mather wrote an unpublished manuscript giving a minimal proof of the classification of the germs of functions that give rise to elementary catastrophes.
• The basic idea localizes functions to germs, then uses determinacy to reduce germs to jets, and reduce the ∞-dimensional problem in analysis to a finite dimensional problem in algebraic geometry.
• Mather paper is confined to the local problem of classifying germs of functions.
• Three steps are needed to put the theory in a usable form for applications.
• Need to globalize from germs back to functions.
• Need Thom transversality lemma
• Chapter 8 is based on Levine's exposition [2].
• The function germs, as classified by Mather, need to be related to induced elementary catastrophes (needed for the applications).
• The elliptic umbilic starts as an unstable germ R^2 -> R, unfolds to a stable-germ R^2 x R^3 -> R, and induces the elementary catastrophe germ Xf: R^3 -> R.
• Chapter 9 verifies the stability of the elementary catastrophes
• Elementary catastrophes are singularities, and stable, they differ from the stable-singularities.
• The unfolded germ is indeed a stable-singularity, but the induced catastrophe germ may not be.
• M denotes the space of all C∞ maps R^r -> R, C the subspace of catastrophe maps.
• C ⊄ M because not all maps can be induced by a function.
• A stable-singularity may appear in M but not in C, and therefore will not occur as an elementary catastrophe.
• An elementary catastrophe may appear in C, and be stable in C, but unstable if perturbations in M are allowed, and therefore will not occur as a stable-singularity.
• For r = 2 the two concepts accidentally coincide because only the fold and cusp are the two stable-singularities and elementary catastrophes.
• For r = 3 the concepts diverge, and for r = 4, there are 6 stable-singularities and 7 elementary catastrophes.
• The classification becomes infinite for r > 5
• Reason for keeping r≤ 5
• There are equivalence classes of singularities depending upon a continuous parameter.
• A finite classification is possible under topological equivalence, but applications need smooth classification in low dimensions.
• The theorem remains true when R^(n+r) is replaced by a bundle over an arbitrary r-manifold, with fibre an arbitrary n-manifold.
• A classification theorem classifies the types of singularity that most Xf can have.
• If Xf has singularity at (x,y) ∈ R^(n+r) ∩ Mf, then the equivalence class of Xf depends on the (right) equivalence class of η (Theorem 7.8).
• This requires the Malgrange Preparation Theorem, the Division Theorem (Chapter 5), and the category of unfoldings of a germ η (Chapter 6).

Classifying Germs

• Classify germs η of C∞ functions R,0 -> R,0.
• Use determinacy, codimension, and the jacobian ideal Δ(η).
• Δ(η) is the ideal spanned by ∂η/∂x1, ..., ∂η/∂xn in the local ring E of germs at 0 of C∞ functions R,0 -> R,0.
• The determinacy of a germ η the least integer k such that if any germ ξ has the same k-jet as η then ξ is right equivalent to η.
• Theorem 2.9 gives necessary and sufficient conditions for k-determinacy in terms of Δ.
• The codimension of η is the dimension of m/Δ, where m is the unique maximal ideal of E.
• Theorem states that det η ≥ 2 ≤ cod η in Lemma 3.1.
• If r ≤ 5 and f ∈ F*, then if η = f|Rxy, for any y ∈ R^r, then cod η ≤ r:.
• Since we can restrict to cod η ≤ 5 we need only look at 7-determined germs in the vector space J7 of 7-jets.
• Restrict to r ≤ 5, for if cod η ≥ 7 there are equivalence classes depending upon a continuous parameter.
• The definition of F* ensures that if r = 6 then each of these equivalence contains an f|Rxy for some y ∈ R^r and f ∈ F*.

Codimension and Determinacy

• Chapters discuss determinacy, with the main results on it completed in Chapter 4, and connectedness of the germs and catastrophe germs completed in Chapter 7.
• Finally in Chapter 9 the local stability of 𝑋_𝑓 is shown, i.e. it is stable under small perturbations of f.
• m_n is a maximal ideal of E_n.
• G_n is a group with multiplication induced by composition.
• k is least integer such that n us k-determinate.
• Jk is spaces of k-jets, or jet spaces.
• Tensors say that if they are right equivalent if they belong to the same G-orbit.

Stating Thom's Theorem

Definitions

• Let f : R^n × R^r -> R be a smooth function.
• Define M_f ⊂ R^(n+r) to be given by (∂f/∂x_1,...,∂f/∂x_n = 0), where x_1,...,x_n are coordinates for R_n.
• Let y_1,...,y_r be coordinates for R^r.
• Generically, M_f is an r-manifold because it is codimension n, given by n equations.
• Let X_f : M_f -> R^r be the map induced by projection R^(n+r)->R^r.
• We call 𝑋_f the catastrophe map of f.
• Let ℱ denote the space of C^∞-functions on R^(n+r) with the Whitney C^∞-topology.
• Theorem Given r <= 5, these is an open dense set ℱ in ℱ which we call generic functions.
• Statement 3:

1. Mf is an r-manifold.
2. Any singularity of 𝑋f is equivalent to one of a finite number of types called elementary catastrophes.
3. 𝑋f is locally stable at all points of Mf with respect to small perturbations of f_

Equivalence of Maps

• Here equivalence means the following:
• Two maps 𝑋 : M -> N and 𝑋' : M' -> N' are equivalent if exist diffeomorphisms h,k such that the following diagram commutes: (diagram in notes).
• Note this means now suppose the maps 𝑋, 𝑋' have singularities at x, x' respectively, then the singularities are equivalent if this previous definition holds locally, i.e. hx=x'.

This quiz covers the key concepts related to catastrophe theory, including the distinctions between elementary catastrophes and stable-singularities and Whitney's findings. It also covers elliptic umbilics, and fundamental components such as the germs x1 and ideals m^k.