| Week |
Date of Monday |
Topics |
Sections in Reid-Szendrői |
Example sheet |
Comments |
| 1 |
Sep. 29 |
Introduction. Metric spaces and examples. Isometric
embeddings and isometries. The Cauchy-Schwarz inequality. |
Appendix A, 1.1 |
One |
See the lovely first chapter of Steele's book
for several proofs of the Cauchy-Schwarz inequality.
|
| 2 |
Oct. 6 |
Platonic solids and \(S^2\). The triangle inequality in
\(\EE^n\). Distance versus arc-length. Lines, collinearity,
parallelity, and Playfair's axiom. Axiomatic systems and models
of such. |
1.2 - 1.4 |
Two |
All five of the Platonic
solids can be inscribed in \(S^2\), after rescaling.
|
| 3 |
Oct. 13 |
Barycentric coordinates. Isometries preserve barycentric
coordinates, are affine maps. Angles. Polarization identity.
Isometries preserve angles. Orthonormal bases, orthogonal
matrices. Orientation preserving/reversing. Rotations and
reflections in \(\EE^2\). |
1.5 - 1.11.1, Appendix B.1 |
Three |
|
| 4 |
Oct. 20 |
Orthogonal complements and the classification of orthogonal
matrices in all dimensions. The parallel postulate and the angle
sum of a euclidean triangle. Classification of isometries of
\(\EE^2\). The cosine law. The sine law and its connection to
area. |
1.11.2 - 1.16.3 (2.1 - 2.6) |
Four |
Please read sections 2.1 - 2.6, on composition of
isometries. |
| 5 |
Oct. 27 |
Spherical distance, great circles, polar duals, and angles.
The spherical cosine law and triangle inequality. Classification
of isometries of the two-sphere. |
3.1 - 3.4 |
Five |
|
| 6 |
Nov. 3 |
Spherical area, lunes, and Girard's lemma. Hyperbolic trig,
Lorentz dot product, the hyperbolic plane. Hyperbolic distance,
angles, unit tangent vectors. |
3.5 - 3.9 |
Six |
|
| 7 |
Nov. 10 |
Lorentzian orthogonals, lines parametrized by arclength,
characterizing unit tangents. Cosine law and the triangle
inequality. Semidirect products, \(\Isom(\EE^2) \isom \RR^2 \semi
O(2)\), \(\Isom(S^2) \isom O(3)\). |
3.10 - 3.11 |
Seven |
|
| 8 |
Nov. 17 |
\(\Isom(\HH^2) \isom O^+(1,2)\). Rotations, translations,
parabolics. Pencils in the three geometries. Elliptic,
parabolic, hyperbolic pencils. Klein model. Independence of the
parallel postulate. |
3.12 - 3.13 |
Eight |
|
| 9 |
Nov. 24 |
Area of hyperbolic triangles. Ideal triangles. Affine space,
subspaces, transformations. The dimension formula, the affine
group. Klein's Erlangen program. Projective equivalence,
projective space. |
3.14, 4.2 - 4.6, 6.3, 5.2 |
Nine |
|
| 10 |
Dec. 1 |
Projective line, projective plane. Dimension formula, the
projective group. Three-transitivity. Desargues's theorem. |
5.3 - 5.11 |
Ten |
|
