Geometry III/IV

Michaelmas 2018

Textbooks:

• V. V. Prasolov, Non-Euclidean Geometry ---- (this book was distributed in the class, thanks to the kind permission of the author and the publisher).
• G. Jones, Algebra and Geometry ---------------- (this lecture notes are available on DUO, thanks to the kind permission of the author).

Further reading and some on-line resources on geometry

Who is who

Preliminary course content for Michaelmas (subject to change):
Euclidean geometry, spherical geometry, affine and projective geometries

Schedule:

• 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: Hyperbolic geometry: Klein disc model (distance, isometries, perpendicular lines).
• Week 10: Classical theorems: Pappus and Desargues. Projective plane: its topology, polarity on projective plane. Problems class on projective geometry.

Homeworks:
- There will be 4 sets of marked homework assignments per term (to hand in on Fridays, weeks 3,5,7,9). -- (+/- notation used for marking)
- There will be also weekly unmarked sets of exercises. Please solve them timely!
- In addition, there will be additional reading material for students that are enrolled in MATH4141 (Geometry IV), -- here is the description.

• Weeks 1-2: ---- Exercises ---- Hints ---- ------ ------ (Isometries of Euclidean plane)
• Weeks 3-4: ---- Exercises ---- Hints ---- ------ ------ (Isometries; actions of groups; a bit of spherical geometry)
• Weeks 5-6: ---- Exercises ---- Hints ---- ------ ------ (Spherical geometry)
• Weeks 7-8: ---- Exercises ---- Hints ---- ------ ------ (Affine and Projective geometry)
• Weeks 9-10: -. Exercises ---- Hints ---- ------ ----- (Projective geometry, Klein model of hyperbolic geometry)

Outlines:

• Outline
• Questions discussed in Problems Classes

Handouts:

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

• 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