To give an introduction to the study of local singularities of differentiable functions and mappings.

(LO1) Operate with the notion of a manifold.

(LO2) Detect stable singularities of maps between low-dimensional manifolds.

(LO3) Apply Thom’s Strong Transversality Theorem in essential situations.

(LO4) Apply various methods for normal form reduction of function germs in two variables.

(S1) Problem solving skills

(S2) Numeracy

(S3) Adaptability

Inverse and implicit function theorems; Morse Lemma;

Manifolds; tangent bundles; vector fields;

Germs of functions and mappings;

Derivative of a mapping between manifolds;

Critical points and critical values of mappings; Sard's lemma.

Equivalence of map-germs; stable map-germs of a plane into a plane; transversality; jet spaces; Thom's transversality theorem.

Local algebra of a singularity; local multiplicity of a mapping; Preparation theorem.

Stability and infinitesimal stability; finite determinacy; versal deformations of functions.

Beginning of the classification of function singularities; Newton diagram; ruler rotation method; simple functions; boundary function singularities.