MATH45051 - 2010/2011
- Title: Singularities, Bifurcations and Catastrophes
- Unit code: MATH45051
- Credit rating: 15
- Level: 3
- Pre-requisite units: MATH20201 Algebraic Structures I, MATH20132 Calculus of several variables.
- Co-requisite units: None
- School responsible: Mathematics
- Members of staff responsible: Dr. J. Montaldi
This lecture course aims to introduce students to the fundamental ideas of classifying and understanding singularities of functions, and to enable them to apply these notions to bifurcation problems.
Singularity theory is the study of the local structure of maps Rn to Rp (or Cn → Cp), particularly when the conditions of the inverse or implicit function theorems fail, and how this structure changes as the map is deformed. In this course we will concentrate on the scalar case (p=1), with the level 4 version including cases with p>1. Such questions arise in many applications, particularly in bifurcation theory and what is sometimes called catastrophe theory.
Intended learning outcomesOn successful completion of this course unit students will be able to:
- use algebraic techniques to compute the codimension of a singularity or a bifurcation;
- classify degenerate critical points;
- find a versal unfolding of a singularity;
- apply the theory of unfoldings to study bifurcations in physical systems.
Future topics requiring this course unit
- Families of functions: Critical points, Hessian and (non)degeneracy, Morse lemma, catastrophe set, discriminant, Singular points of maps, families of maps. 
- Changes of coordinates and submanifolds: diffeomorphisms, inverse function theorem, germs, implicit function theorem, submanifolds, parametrization. Linearly adapted coordinates, transversality, Lyapunov-Schmidt reduction. 
- Some algebra: ring of germs of smooth functions, maximal ideal, Nakayama's lemma, Newton diagram, finite codimension ideals. Filtrations, modules over rings of polynomials. 
- Critical points: Jacobian ideals, codimension, right equivalence, finite determinacy, splitting lemma, classification of critical points. Other singularity-theoretic equivalences, their tangent spaces, codimension and finite determinacy. 
- Unfoldings: Families of functions as unfoldings, versal unfoldings, versality theorem for right-equivalence. Versality for other equivalences. 
- Applications: Illustration of types of application, such as the geometry of curves and surfaces, gravitational catastrophe machine, ship stability, or other examples. Bifurcation of equilibria, pitchfork bifurcation, Hopf bifurcation 
- V. Arnold, V. Goryunov, O.V. Lyashko & V.A. Vasil'ev, Singularity Theory I. Springer. 1993.
- C.G. Gibson, Singular Points of Smooth Mappings. Pitman, 1979.
- T. Poston and I.N. Stewart, Catastrophe Theory and its Applications. Dover.
- Th. Bröcker, Differentiable Germs and Catastrophes, LMS Lecture Notes, CUP.
Learning and teaching processes
Two lectures and one examples class each week, supplemented with extra reading provided by the lecturer plus 4 or 5 extra lectures. In addition students should expect to spend at least seven hours each week on private study for this course unit.
- Mid-semester test: weighting 10%
- End of semester examination: three hours, weighting 90%