\( \def\Fix{\mathrm{Fix}} \def\D{\mathsf{D}} \def\C{\mathsf{C}} \def\D{\mathsf{D}} \def\GL{\mathsf{GL}} \def\OO{\mathsf{O}} \def\SO{\mathsf{SO}} \def\RR{\mathbb{R}} \def\ZZ{\mathbb{Z}} \def\Aut{\mathrm{Aut}} \def\xx{\mathbf{x}} \)
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]

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