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.
- Thom-Boardman classification.
- Index of a singular point of a mapping Rn → Rn and of a smooth function. Global index and Euler
- 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
- 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.