MAGIC084: Singularity Theory

Course details


Autumn 2013
Monday, October 7th to Saturday, December 14th


Live lecture hours
Recorded lecture hours
Total advised study hours


10:05 - 10:55


The course is an introduction to Singularity Theory, which is also known as Catastrophe Theory. It provides tools to study sharp changes, bifurcations and metamorphoses taking place in various systems depending on parameters under continuous changes of these parameters. You will become familiar with basic notions and theorems used in Singularities. A technical part of the course will be devoted to reduction of functions to local normal forms, which is a far-reaching generalisation of the classification of extrema of functions well-known from school.


There are no prerequisites beyond a standard undergraduate curriculum: elements of group theory, linear algebra, real and complex analysis. Some knowledge of differentiable manifolds, Lie groups and Lie algebras would be helpful but is not compulsory.


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. Finite determinacy, Tougeron’s theorem; versal deformations of functions. Beginning of the classification of function singularities; Newton diagram; quasihomogeneous and semi-quasihomogeneous functions; ruler rotation method; simple functions; Arnold’s spectral sequence; boundary function singularities.


  • VG

    Prof Victor Goryunov

    University of Liverpool


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The assessment for this course will be via a single take-home paper in January and with 2 weeks to complete and submit online. The exam paper will consist of three questions. All questions will carry equal weight and each question will be marked out of total 25 points. To pass the exam you will need at least 40 points out of the total 75 points.

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