Curves and Singularities (MAGIC027) |
GeneralDescription
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.
SemesterSpring 2010 (Monday, January 11 to Friday, March 19) Timetable
Prerequisites
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!
Syllabus
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.
Lecturer
Students
Bibliography
Note: 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.) AssessmentNo assessment information is available yet.
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