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 Prof VV Goryunov
Mathematical Sciences
Victor.Goryunov@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2020-21 M Level 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

(S3) Adaptability

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.

Recommended Texts
Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.

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

Co-requisite modules:
 

Modules for which this module is a pre-requisite:
 

Programme(s) (including Year of Study) to which this module is available on a required basis:
 

Programme(s) (including Year of Study) to which this module is available on an optional basis:
 

Assessment
EXAM Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
final assessment open book and remote standard UoL penalty applies for late submission  one hour time on tas    40       
CONTINUOUS Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
Homework 3 Standard UoL penalty applies for late submissions  equivalent to 2-5 si    20       
Homework 2 Standard UoL penalty applies for late submissions  equivalent to 2-5 si    20       
Homework 1 Standard UoL penalty applies for late submissions  equivalent to 2-5 si    20