Geometry III/IV
Michaelmas 2016 and Epiphany 2017
- 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 (subject to change):
Euclidean geometry, spherical geometry, affine and projective geometries, Möbius transformations, hyperbolic geometry, further topics (geometric surfaces, discrete groups).
Schedule:
- Week 1: Introduction. Axioms: Euclid and Hilbert.
- Week 2: Euclidean geometry: isometry group, its generators, conjugacy classes.
- Week 3: Problems class on geometric constructions and on using reflections for solving problems. 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.
- Week 4: Euclidean geometry in 3 dimensions. Spherical geometry: distance, triangle inequality, geodesics.
- Week 5: Problems class on discrete actions. Spherical geometry: polar correspondence,
congruence of triangles, sine and cosine rules.
- Week 6: Spherical geometry: area of a triangle; isometries on the sphere.
- Week 7: Problems class on spherical geometry. Affine geometry. Projective line.
- Week 8: Projective line and projective plane.
- Week 9: Problems class on projective geometry.
Projective plane: its topology, polarity on
projective plane. Classical theorems: Pappus and Desargues.
- Week 10: Hyperbolic geometry: Klein disc model (distance, isometries, perpendicular lines).
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- Week 11: Möbius transformations, Inversion.
- Week 12: Problems class on Inversions and Möbius geometry.
Möbius transformations and cross-ratios. Inversion in space and stereographic projection.
- Week 13: Conformal models of hyperbolic geometry (Poincare disc).
- Week 14: Problems class on Poincare disc model. Conformal models of hyperbolic geometry (upper half-plane model).
- Week 15: Elementary hyperbolic geometry: sine and cosine rules, area of a triangle.
- Week 16: Projective models of hyperbolic geometry: Klein model and hyperboloid model.
Problems class: area of a disc, hyperbolic oranges; constructions with hyperbolic ruler and compass and other things.
- Week 17: Types of isometries of the hyperbolic plane. Horocycles and equidistant curves.
- Week 18: Taming infinities with horocycles. Family of geometries: sphere-plane-hyperbolic plane. Examples of discrete groups acting on the hyperbolic plane. Problems class: computations in the Klein model.
- Week 19: Hyperbolic surfaces. Review via 3D hyperbolic geometry.
If you have any questions you are very welcome to ask (during the lectures, after a lecture,
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 Thursdays, weeks 3,5,7,9 and 13,15,17,19). --
(+/- 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)
- Christmas problems ---- ----------------------. (additional problems, not compulsory)
- Weeks 11-12: Exercises ---- Hints ---- ------- ------ (Möbius geometry, Inversion)
- Weeks 13-14: Exercises ---- Hints ---- ------- ------ (Poincare disc, upper half-plane)
- Weeks 15-16: Exercises ---- Hints ---- ------- ------- (Computations on the upper half-plane. Klein disc and hyperboloid models)
- Weeks 17-18: Exercises ---- Hints ---- ------- ------- (Classification of isometries. Horocycles and equidistant curves)
Outlines:
Tables:
Other handouts:
If you find mistakes/misprints in solutions or handouts, please let me know - Thanks!