# Symmetry in Nature

**MATH35081**

Lecturer: Dr James Montaldi (Office Hour: Mondays 12:30 - 13:30)

Schedule (2017-18): Wednesday 12-1 (Zochonis Theatre B) and Friday 12-2 (Ellen Wilkinson C5.1).

*Office hour:* Tuesdays 1:30 - 2:30

### Pre-requisite

A first course in Group Theory, such as MATH20201

### Aims

To develop an understanding of symmetry as it arises in nature, and to develop the mathematical techniques for its study through the action of groups.

### Assessment Methods

- Coursework (worth 20%) set around the middle of the semester
- End of semester examination (worth 80%).

### Feedback

Feedback for this course is given in two ways: at weekly problems classes and on the take-home coursework, which will be returned in Week 10. Students can also obtain feedback on their understanding of the material in the course by directly asking the lecturer, either following a lecture or in the lecturer's office hour.

### Syllabus

This is the unofficial syllabus (the descriptions might be modified as we progress): the official one is here

**What is symmetry?**Examples. Group actions. Orbits and stabilizers. Action on \(G/H\). [5 lectures]**Symmetry in geometry:**Example - classification of triangles**Classification:**of symmetry groups in 2 and 3 dimensions**Symmetry of lattices:**(frieze patterns, wallpaper groups and crystals)**Symmetry and ODEs:**symmetric and non-symmetric solutions; spontaneous symmetry breaking**Spatio-temporal symmetry:**(ie, symmetries of periodic orbits, eg coupled cells)**Symmetry and PDEs:**pattern formation and more spatio-temporal symmetry [time permitting - not covered in 2016-17]

### Recommended Reading

*General:*

- I.N. Stewart,
*Symmetry, a very short introduction*, Oxford (2013) - H. Weyl
*Symmetry*, Princeton Science Library (1952)

*Mathematical:*

- M.A. Armstrong,
*Groups and Symmetry*, Springer (1997)

(this book is an introduction to Group Theory, using symmetry as its motivation.)

*Advanced:*

- M. Golubitsky & I. Stewart,
*The Symmetry Perspective*, Birkhauser Verlag (2002) - R. Hoyle,
*Pattern Formation*, CUP (2006)

### Study Hours

- Lectures - 22 hours
- Tutorials - 11 hours
- Independent study hours - 67 hours