### Module Details

 The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
 Title Singularity Theory of Differentiable Mappings Code MATH455 Coordinator Professor VV Goryunov Mathematical Sciences Victor.Goryunov@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2023-24 Level 7 FHEQ First Semester 15

### Aims

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

### Learning Outcomes

(LO1) To know and be able to apply the technique of reducing functions to local normal forms.

(LO2) To understand the concept of stability of mappings and its applications.

(LO3) To be able to construct versal deformations of isolated function singularities.

(S1) Problem solving skills

(S2) Numeracy

### Syllabus

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.

### Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH101 Calculus I 2020-21; MATH102 CALCULUS II 2020-21; MATH103 Introduction to Linear Algebra 2020-21

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
final assessment  120    70
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Homework 1 Standard UoL penalty applies for late submissions    10
Homework 2 Standard UoL penalty applies for late submissions    10
Homework 3 Standard UoL penalty applies for late submissions    10