Mathematical Catastrophe Theory

Prerequisites: Multivariable Calculus; Linear Algebra; Algebra: algebras, ideals, quotient algebras.

Curriculum:

  1. Critical and noncritical points. Implicit function theorem.
  2. Morse lemma. Examples of non-Morse functions.
  3. Typical singularities of mappings R2 → R2 and R2 → R3. Equivalence of singularities. Stable singularities.
  4. Thom-Boardman classification.
  5. Index of a singular point of a mapping Rn → Rn and of a smooth function. Global index and Euler characteristic.
  6. Calculus of jets. Transversality theorems. Whitney (weak) embedding and immersion theorems.
  7. Multiplicity of a smooth map Cn → Cn. Milnor number of a smooth function (geometrical definitions).
  8. Rings of polynomials and power series. Their ideals. Local algebra of a singularity. Sufficient jet theorem. The Milnor number as the dimension of the local algebra.
  9. Newton polyhedra and their applications.
  10. Classification of singular points of functions and their normal forms. Simple singularities.
  11. Milnor fiber and its topology. Monodromy operator.
  12. Braid groups. Deformations of singularities, monodromy groups. Galois group of a general polynomial equation of degree n. Applications to complexity theory. Applications to problems of integral geometry. Simple singularities and reflection groups.
  13. Resolution of singularities of algebraic varieties.