**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.