Summary - week by week

Lectures in green have not happened yet

Chapter 1

Week 1. What is symmetry? Transformations and groups. And a reminder about permutations. Definition of group actions. Orbits and stabilizers.

Week 2. More on orbits and stabilizers. Orbit-Stabilizer theorem. Free, transitive and effective actions. Three actions of a group on itself.

Week 3. Action on set of left cosets \(G/H\) as model for any transitive action. Burnside type of action.

Chapter 2

Week 4. Euclidean transformations, \(\OO(n)\) and \(\SO(n)\); Seitz symbol. Elements of \(\OO(2)\) are reflections or rotations.

Week 5. Finite subgroups of \(\OO(2)\): The cyclic subgroup \(C_n\) and the dihedral subgroups of \(\OO(2)\). Euclidean transformations in the plane: translations, rotations, reflections and glide reflections. – can now do all problems up to 2.20

Week 6 (start). Euclidean transformations and classification of triangles. – can now do all problems of Chapter 2

Chapter 3

Week 6 (ctd). Lattices. The 5 types of lattice in \(\RR^2\) - can do problems 3.1-3.5, 3.7 & 3.10

Week 7. Symmetry groups of lattices. Wallpaper groups. Can do all problems of Chapter 3

Chapter 4

Week 8. Symmetric problems and invariant functions . Critical points of invariant functions and Fixed point subspaces. Principle of symmetric criticality.

Week 9. Bifurcations and symmetry breaking, axial subgroups and tetrahedral example. Symmetric ODEs (start of Chapter 5)

- - Easter Break - -

Chapter 5

Week 10. Symmetry in differential equations and conservation of symmetry (the Symmetry Principle in action). Examples - coupled cell systems

Chapter 6

Week 11. Periodic motion and spatio-temporal symmetry

Fri May 8th is a holiday