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

Summary - lecture by lecture

Lectures in green have not happened yet

Chapter 1

1. What is symmetry? Transformations and groups. And a reminder about permutations.

2. Group actions. Orbits and stabilizers.

3. More on orbits and stabilizers.

4. Orbit-Stabilizer theorem. Free, transitive and effective actions.

5. Actions of a group on itself.

6. Action on set of left cosets \(G/H\) as model for any transitive action. Burnside type of action.

You should now be able to do all problems from Chapter 1, except for 1.16 and 1.18. Those two problems require Sec 1.6 (which isn't covered in lectures and isn't examinable at the final exam). On the other hand, the coursework will be testing that section.

Chapter 2

7. Euclidean transformations, \(\OO(n)\) and \(\SO(n)\); Seitz symbol

8. Elements of \(O(2)\) are reflections or rotations. The subgroup \(C_n\)

9. Dihedral subgroups of \(\OO(2)\). The finite subgroups of \(\OO(2)\). Euclidean transformations in the plane: rotations, . . .

10. . . . reflections and glide-reflections. Euclidean transformations and classification of triangles.

11a. Finish classification of triangles. can now do all problems of Chapter 2

Chapter 3

11b. Lattices

12. The 5 types of lattice in \(\RR^2\);

13. Symmetry groups of lattices. Can do problems 3.1 to 3.12

14. Wallpaper groups. Can do all problems of Chapter 3

Chapter 4

15. Symmetric problems and invariant functions can do problem 4.1

16. Critical points of invariant functions and Fixed point subspaces

17. Principle of symmetric criticality

18. Bifurcations and symmetry breaking;

19. Example in 3D: tetrahedral group

Chapter 5

20. Symmetry in differential equations

21. Conservation of symmetry and coupled cell systems

Chapter 6

22. Symmetries of periodic orbits

23. Revision discussion