Prerequisites: Multivariable Calculus; Linear Algebra; Algebra: algebras, ideals, quotient algebras.
- Critical and noncritical points. Implicit function theorem.
- Morse lemma. Examples of non-Morse functions.
- Typical singularities of mappings R2 → R2 and R2 → R3. Equivalence of singularities. Stable singularities.
- Thom-Boardman classification.
- Index of a singular point of a mapping Rn → Rn and of a smooth function. Global index and Euler characteristic.
- Calculus of jets. Transversality theorems. Whitney (weak) embedding and immersion theorems.
- Multiplicity of a smooth map Cn → Cn. Milnor number of a smooth function (geometrical definitions).
- 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.
- Newton polyhedra and their applications.
- Classification of singular points of functions and their normal forms. Simple singularities.
- Milnor fiber and its topology. Monodromy operator.
- 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.
- Resolution of singularities of algebraic varieties.