# MMath Mathematics with Financial Mathematics

Year of entry: 2023

## Course unit details:Symmetry in Geometry and Nature

Unit code MATH35082 10 Level 3 Semester 2 Department of Mathematics No

### Overview

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.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Algebraic Structures 1 MATH20201 Pre-Requisite Compulsory

### Aims

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.

### Learning outcomes

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.

### Syllabus

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.

### Assessment methods

Method Weight
Other 20%
Written exam 80%

Take-home coursework and online tests (if appropriate) in total worth 20%.

End of semester examination (worth 80%).

### Feedback methods

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.

General:
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)

### Study hours

Scheduled activity hours
Lectures 12
Tutorials 12
Independent study hours
Independent study 76

### Teaching staff

Staff member Role
James Montaldi Unit coordinator

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

•         You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently
•         You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week
•         There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises
•         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.
The remaining independent study time comprises revision for and taking the end-of-semester assessment.

The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.