Geometry III/IV
Michaelmas 2018
- 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).
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.
If you have any questions you are very welcome to ask (during the lectures, after a lecture,
during office hours, in any other convinient time or via e-mail)!!!
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:
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:
If you find any mistakes/misprints in solutions or handouts, please let me know - Thanks!