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.
• Week 13: Conformal models of hyperbolic geometry (Poincaré disc). Problems class on Inversions and Möbius geometry.
• Week 14: Isometries of Poincaré disc. Circles in Poincaré disc. Upper half-plane model.
• Week 15: Elementary hyperbolic geometry: sine and cosine rules. Problems class on Poincaré disc.
• Week 16: Area of a triangle. Projective models of hyperbolic geometry: Klein model. Hyperboloid model.
• Week 17: Hyperboloid model. Problems class: area of a disc, hyperbolic oranges, playing hyperbolic sports.
• Week 18: Types of isometries of hyperbolic plane. Horocycles and equidistant curves. Discrete reflection groups.
• Week 19 Taming infinities with horocycles. Family of geometries: sphere-plane-hyperbolic plane. Discrete reflection groups. Problems class: computations in the Klein model.
• Week 20: Hyperbolic surfaces. Review via 3D hyperbolic geometry.

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	Solutions	Feedback	Möbius geometry, inversion
13-14	HW 13-14	Hints 13-14	February 21	Solutions	Feedback	Poincare disc, upper half-plane
15-16	HW 15-16	Hints 15-16	March 7	Solutions	Feedback	Computations in UHP, Klein disc and hyperboloid models
17-18	HW 17-18	Hints 17-18	March 21	Solutions	Feedback	Classification of isometries. Horocycles and equidistant curves

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.

Some tables to review the course:

• Formulae sheet: you will get a copy during the exam
• Models of hyperbolic geometry: overview
• Groups in Geometries: overview
• Spherical, Euclidean and hyperbolic geometries: comparison

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.