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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.


Autumn 2015 (Monday, October 5 to Friday, December 11)


  • Wed 13:05 - 13:55


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.


Victor Goryunov
Phone (0151) 7944041
Interests Singularity theory, Vassiliev invariants


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Zhe Chen
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Alessio Cipriani
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Maxime Fairon
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Calum Horrobin
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Stephen Nand Lal
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Konstantinos Palapanidis
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Bogna Pawlowska
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Pedro Peres
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Sam Povall
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Courtney Quinn
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Marzia Romano


Modern Birkh‰User Classics: Singularities of Differentiable Maps ...Arnold, Gusein-Zade and Varchenko
Singularity Theory. I. Reprint of the original English edition from the series Encyclopaedia of Mathematical SciencesArnold, Goryunov, Layshko
Curves and Singularities: A Geometrical Introduction to Singularity TheoryBruce and Giblin
Stable Mappings and Their Singularities [By] M. Golubitsky [And] V. GuilleminGolubitsky and Author)


Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)


The assessment for this course will be via a single take-home paper in January with 2 weeks to complete and submit online. The paper will have three questions, each worth 25 points. To pass the exam you will need 40 points out of the total 75.

Singularity Theory

Files:Exam paper
Released: Monday 4 January 2016 (628.6 days ago)
Deadline: Friday 15 January 2016 (616.6 days ago)

Recorded Lectures

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