MAGIC027: Curves and Singularities

Course details

A specialist MAGIC course


Spring 2009
Monday, January 19th to Friday, March 27th; Monday, April 27th to Monday, April 27th


Live lecture hours
Recorded lecture hours
Total advised study hours


11:05 - 11:55


There are two Mondays when I shall need to do some rearrangement.
On Monday 9 March I want to go to a meeting at Warwick on computer vision (the area to which I apply singularity theory).
If possible I will do a swap with someone else that week, and this will be announced well in advance.
On Monday 23 March I expect to be in Valencia at a workshop 'Singularities in Generic Geometry and Applications'.
Instead of 23 March I could give an extra lecture the following Monday, 30 March, if that would be convenient.


Welcome to Curves and Singularities. The topic of this course is really 'singularities of functions of 1 variable and their unfoldings'; it is intended to be a concrete introduction to the ideas of modern singularity theory, using curves, families of curves and families of surfaces (in 3-space) as the geometrical material whose properties can be found using singularity theory. A singularity of a function is just a 'turning point' and for a function of one variable we can measure just how singular a function is by counting the number of derivatives which vanish at a particular value of the variable. Even this simple idea has enormous geometrical implications which we shall explore. Similar ideas using two or more variable allow the study of the geometry of surfaces by means of singularities of functions and mappings. These methods go back to Whitney and Thom in the 1950s and 1960s but they are still a very active research area today.
Apart from its applications within mathematics, singularity theory has many applications outside, for example in computer vision (my own area of application). To convince yourself of this, try typing some of these keywords into Google: medial axis, symmetry set, ridge curve, apparent contour.


The course uses basic calculus and a little linear algebra. The main prerequisite is a wish to understand better the geometry of curves and surfaces in euclidean space!


Curves, and functions on them. Classification of functions of 1 real variable up to R-equivalence. Regular values of smooth maps, manifolds. Applications. Envelopes of curves and surfaces. Unfoldings of functions of 1 variable. Criteria for versal unfolding.


  • PG

    Professor Peter Giblin

    University of Liverpool


No bibliography has been specified for this course.


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