Geometry III/IV

Michaelmas 2016 and Epiphany 2017

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 (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.

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:

• Outline
• Questions discussed in Problems Classes

Tables:

• 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

Other handouts:

• On relations between inversions and Möbius transformations.
• Example: how a proof in a model goes.