Teaching methods (Werkvormen)
- 28 hrs lecture
- 28 hrs problem session
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Pre-requisites (Vereiste voorkennis)
Symmetry, Topology
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Contents (Beschrijving)
Algebraic topology is concerned with algebraic characteristics (numbers, groups, rings, etc) of spaces which provide information about their geometric properties. The most well known such algebraic data are the homotopy groups of a space, and its homology and cohomology groups. First examples of such geometric properties easily proved with the methods of algebraic topology are the following two: If you have a dog which is completely covered in hair, then there is no way of combing that hair smoothly, so that there is no parting or bald spot. This is the so-called `hairy dog theorem'. If you associate to each point of the Earth's surface the two numbers t and p given by temperature and air pressure at this point, then there is always at least one point which has the same values for t and p as its diametrically opposite one. This is the so-called 'Borsuk-Ulam theorem'. The (co)homological methods of algebraic topology have later been used in many other parts of mathematics, for example in algebraic geometry and group theory, and are now a standard tool in these subjects. Thus, the course will be of interest to a wide variety of master students.
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