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

### Curriculum:

- Critical and noncritical points. Implicit function theorem.
- Morse lemma. Examples of non-Morse functions.
- Typical singularities of mappings
**R**^{2}→**R**^{2}and**R**^{2}→**R**^{3}.

Equivalence of singularities.

Stable singularities. - Thom-Boardman classification.
- Index of a singular point of a mapping
**R**^{n}→**R**^{n}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
**C**^{n}→**C**^{n}.

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.