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BSc Actuarial Science and Mathematics / Course details
Year of entry: 2021
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Course unit details:
Symmetry in Geometry and Nature
|Unit level||Level 3|
|Teaching period(s)||Semester 2|
|Offered by||Department of Mathematics|
|Available as a free choice unit?||No|
Symmetry arises frequently, in Nature and in Mathematical models of Nature, and the appreciation of symmetry is deeply ingrained in our consciousness and in our sense of beauty. But symmetry also arises in Mathematics in ways that do not arise from Nature, such as in Galois’ analysis of polynomial equations arising from symmetries in their roots. We will start the course by understanding why Group Theory is the natural mathematical language for symmetry, and study some symmetries arising in geometry, such as the symmetry of a cube, consisting of certain rotations and reflections in space which together form a group. We will discuss the classification of symmetries of repeated patterns, like those in the famous Alhambra Mosque in Grenada, giving rise to the Wallpaper Groups. The second half of the course will look more closely at applications of group actions to ordinary differential equations.
|Unit title||Unit code||Requirement type||Description|
|Algebraic Structures 1||MATH20201||Pre-Requisite||Compulsory|
To develop an understanding of symmetry as it arises both mathematically in Geometry and in Nature, and to develop the mathematical techniques for its study through the action of groups.
On successfully completing the course, the student will be able to:
- apply the orbit-stabilizer theorem,
- determine the Burnside type of a finite group action,
- perform calculations using the Seitz symbol for a Euclidean transformation,
- recognize the 5 types of symmetry of a plane lattice, and use this to analyze wallpaper groups,
- find the symmetric solutions to problems with symmetry,
- predict the possible symmetries of periodic motions in symmetric systems.
1. What is symmetry? Examples. Groups of transformations. Orbits and stabilizers. Orbit types and Burnside type of an action.
2. Euclidean transformations, the Seitz symbol, example of classification of triangles.
3. Classification of symmetry groups in 2 dimensions: point-groups, lattices and wallpa- per groups.
4. Symmetry and ODEs: symmetric and nonsymmetric solutions; spontaneous symmetry breaking.
5. Spatiotemporal symmetry in periodic motion.
Take-home coursework and online tests (if appropriate) in total worth 20%.
End of semester examination (worth 80%).
Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
I.N. Stewart, Symmetry, a very short introduction, Oxford (2013)
H. Weyl Symmetry, Princeton Science Library (1952)
M. Golubitsky & I. Stewart, The Symmetry Perspective, Birkhauser Verlag (2002)
|Scheduled activity hours|
|Independent study hours|
|James Montaldi||Unit coordinator|
This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.
Please see Blackboard / course unit related emails for any further updates