$$\def\Fix{\mathrm{Fix}} \def\D{\mathsf{D}} \def\C{\mathsf{C}} \def\D{\mathsf{D}} \def\GL{\mathsf{GL}} \def\OO{\mathsf{O}} \def\SO{\mathsf{SO}} \def\RR{\mathbb{R}} \def\ZZ{\mathbb{Z}} \def\Aut{\mathrm{Aut}} \def\xx{\mathbf{x}}$$
jam - wiki

# 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

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

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.)