Geometry III/V

2024/25

Textbook:

• V. V. Prasolov, V. M. Tikhomirov, Geometry, American Maths. Soc., 2001. (the e-book is available on Ultra (see "Reading List")

Further reading and some on-line resources on geometry

Who is who

Preliminary course content (subject to change):
Michaelmas topics: Euclidean geometry, spherical geometry, affine and projective geometries;
Epiphany topics: Möbius transformations, hyperbolic geometry, further topics (geometric surfaces, discrete groups).

Schedule:

Michaelmas:
• Week 1: Introduction. Axioms: Euclid and Hilbert.
• Week 2: Euclidean geometry: isometry group, its generators, conjugacy classes.
• Week 3: More on Euclidean isometry group: fixed points of isometries, conjugacy classes of isometries, orthogonal transformations as isometries preserving the origin. Discrete groups acting on Euclidean plane. Problems class on geometric constructions and on using reflections for solving problems.
• Week 4: Euclidean geometry in 3 dimensions. Spherical geometry: distance, triangle inequality, geodesics.
• Week 5: Spherical geometry: polar correspondence, congruence of triangles, sine and cosine rules. Problems class on discrete actions.
• Week 6: Spherical geometry: area of a triangle; isometries on the sphere.
• Week 7: Affine geometry. Projective line. Problems class on spherical geometry.
• Week 8: Projective line and projective plane.
• Week 9: Classical theorems: Pappus and Desargues. Projective plane: its topology, polarity on projective plane. Problems class on projective geometry.
• Week 10: Hyperbolic geometry: Klein disc model (distance, isometries, perpendicular lines).
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Epiphany:
• Week 11: Möbius transformations, Inversion.
• Week 12: Inversion. Möbius transformations and cross-ratios. Inversion in space and stereographic projection. Problems class on Inversions and Möbius geometry.
• Week 13: Conformal models of hyperbolic geometry (Poincaré disc).

Assignments in Epiphany:
- There will be 4 sets of marked assignments during Epiphany term (to submit through Gradescope on Fridays by 6pm, weeks 14,16,18,20). --
- There will be also weekly unmarked sets of exercises. Please solve them timely!
- In addition, there will be additional reading material for MSc students, see here for the instructions.

Week #	Questions	Hints	Due date: Fr 6pm	Solutions	Feedback
1-2	HW 1-2	Hints 1-2	----	Solutions	Feedback	Isometries of Euclidean plane
3-4	HW 3-4	Hints 3-4	----	Solutions	Feedback	Isometries; actions of groups; a bit of spherical geometry
5-6	HW 5-6	Hints 5-6	----	Solutions	Feedback	Spherical geometry
7-8	HW 7-8	Hints 7-8	----	Solutions	Feedback	Affine and Projective geometry
9-10	HW 9-10	Hints 9-10	----	Solutions		Projective geometry, Klein model of hyperbolic geometry

Week #	Questions	Hints	Due: Fr 6pm	Solutions	Feedback
11-12	HW 11-12	Hints 11-12	February 7			Möbius geometry, inversion
13-14			February 21
15-16			March 7
17-18			March 21

Lecture Notes: (Material to be added during the term)

• Lecture notes
• Problems Classes and Revision lectures: Questions and Solutions

Handouts:

• Euclidean geometry: axioms (page 2) and basic theorems (pages 3-4), as well as the contact details on page 1.

Additional handout:
(not an essential part of the course, it is non-examinable!)

• Here you can find an instruction on how to construct vertices of dodecahedron/icosahedron with spherical ruler and compass.