Critical points of functions (MATH20132), existence and uniqueness of solutions of ordinary differential equations (from MATH10222 or similar), familiarity with the notion of fields and rings (MATH20201),
- Introduction: Families of maps 
- Changes of coordinates and diffeomorphisms: inverse function theorem, implicit function theorem, submanifolds, parametrization. Linearly adapted coordinates, transversality, Lyapunov-Schmidt reduction, germs. 
- Algebra: ring of germs of smooth functions, maximal ideal, Newton diagram, finite codimension ideals. Modules. Nakayama's lemma. 
- Right equivalence: Jacobian ideals/modules and R-tangent space, codimension, finite determinacy, splitting lemma, classification of critical points. 
- Singularities of maps: Other singularity-theoretic equivalences, their tangent spaces, codimension and finite determinacy. 
- Unfoldings: Families of maps as unfoldings, versal unfoldings, versality theorem for right-equivalence and contact-equivalence. 
- 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 in ODEs, idea of Hopf bifurcation.