Prerequisites

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),

Syllabus

  1. Introduction: Families of maps [1]
  2. Changes of coordinates and diffeomorphisms: inverse function theorem, implicit function theorem, submanifolds, parametrization. Linearly adapted coordinates, transversality, Lyapunov-Schmidt reduction, germs. [5]
  3. Algebra: ring of germs of smooth functions, maximal ideal, Newton diagram, finite codimension ideals. Modules. Nakayama's lemma. [3]
  4. Right equivalence: Jacobian ideals/modules and R-tangent space, codimension, finite determinacy, splitting lemma, classification of critical points. [5]
  5. Singularities of maps: Other singularity-theoretic equivalences, their tangent spaces, codimension and finite determinacy. [5]
  6. Unfoldings: Families of maps as unfoldings, versal unfoldings, versality theorem for right-equivalence and contact-equivalence. [6]
  7. 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. [3]