Geometry III/V

Epiphany 2022

Textbook:

• V. V. Prasolov, Non-Euclidean Geometry ---- (the paper copies of this book were distributed in class, thanks to the kind permission of the author and the publisher).

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:

• Week 11: Möbius transformations, Inversion.
• Week 12: Inversion. Möbius transformations and cross-ratios. Inversion in space and stereographic projection.
• Week 13: Stereographic projection. 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 Poincare disc model.
• Week 16: Area of a triangle. Projective models of hyperbolic geometry: Klein model.
• Week 17: Hyperboloid model. Types of isometries of hyperbolic plane. Problems class on computations in hyperbolic geometry.
• Week 18: More on isometries. Horocycles and equidistant curves. Discrete reflection groups.
• Week 19: Taming infinities with horocycles. Family of geometries: sphere-plane-hyperbolic plane. More on discrete groups acting on the hyperbolic plane. Problems class: computations in the Klein model.
• Week 20: Hyperbolic surfaces. Review via 3D hyperbolic geometry.

Assignments:
- There will be 4 sets of marked assignments during the term (to submit through Gradescope on Fridays by 5pm, weeks 14,16,18,20). --
- There will be also weekly unmarked sets of exercises. Please solve them timely!

Michaelmas Assignments: (with feedback based on results from previous years)


• 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 ---- (Supplementary assignment: Projective geometry, Klein model of hyperbolic geometry)

Epiphany Assignments: (to be added here)


• Weeks 11-12: Exercises ---- Hints ----(Möbius geometry, Inversion)
• Weeks 13-14: Exercises ---- Hints ----(Poincare disc, upper half-plane)
• Weeks 15-16: Exercises ---- Hints ----(Computations in the upper half-plane. Klein disc and hyperboloid models)
• Weeks 17-18: Exercises ---- Hints ----(Classification of isometries. Horocycles and equidistant curves)

Lecture Notes: (Epiphany matherial 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