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

Videos, week by week

Resources folder

  • Permutations
  • Cosets

+Appendix

The review class takes place in week n+1

Week 1

  • What is symmetry? (25 mins)

Chapter 1

  • Actions (23 mins)
  • Orbits & stabilizers (22 mins)

Total: 70 mins

Week 2

  • Orbit-stabilizer theorem (24 mins)
  • Transitive, free and effectuve actions (11 mins)
  • Equivariance and equivalent actions (12½ mins)
  • Actions of G on itself (16 mins)

Total: 63½ mins

Week 3

  • Action on \(G/H\) (16½ mins)
  • Orbit types (15 mins)
  • Burnside counting theorem (15 mins)
  • Proof of Burnside's theorem (10 mins)

Total: 56½ mins

Week 4

Chapter 2:Euclidean transformations

  • Eucl. transformations (11½ mins)
  • Seitz symbol (7 mins)
  • Elements of O(2) (17 mins)
  • Finite subgroups of O(2) (14½ mins)
  • Translations & Rotations (7 mins)

Total 57 mins

Week 5

  • Glide-reflections (14 mins)

Chapter 3: Wallpaper groups

  • Intro (11 mins)
  • Lattices (10 mins)
  • Crystallographic Restriction Thm (6 mins)
  • The 5 types of lattice (16½ mins)

Total 57½ mins

Week 6

  • Symmetries of lattices (20½ mins)
  • Wallpaper groups - classification (18½ mins)
  • Wallpaper groups - recognition (13 mins)

Total 52 mins

Week 7

Chapter 4: The Symmetry Principle

  • Symmetry principle (3 mins)
  • Invariant functions (20 mins)
  • Critical Points 1 (13½ mins)
  • Symmetry Lemma (7½ mins)
  • Fixed point subspaces (8 mins)
  • Principle of Symmetric Criticality - statement (10 mins)

Total 62 mins

Week 8

Coursework due on Weds 30th March, 2022

  • Principle of Symmetric Criticality - proof (7 mins)
  • Bifurcations (16½ mins)
  • Bifurcations - examples (12 mins)
  • Symmetry breaking & axial subgroups \(D_4\) example (16 mins)
  • Tetrahedral example (13½ mins)

Total 65 mins

Easter Break

Week 9

Chapter 5: Symmetric ODEs

  • Symmetric ODEs (transformations of solutions) (25 mins)
  • Conservation of Symmetry (fixed point subspaces) (16½ mins)
  • Coupled cell systems (15 mins)
  • Examples (14½ mins)

Total 71 mins

Week 10

Chapter 6: Periodic motion

  • Periodic motion - examples (18 mins)
  • \(S^1\) and spatio-temporal symmetry (13 mins)
  • Spatio-temporal symmetry as a graph (20 mins)

Total 51 mins

Week 11

  • Animal gaits (21 mins)
  • Mechanical systems (23 mins)

Total 44 mins

Online coursework held on Tues 10th/Weds 11th May, 2022 (Week 11)