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

After completing this module students should know:

-- technique of reducing functions to local normal forms;

-- understand the concept of stability of mappings;

-- construct versal deformations of isolated function singularities.

Inverse and implicit function theorems; Morselemma; 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.