Geometry III/V

2025/26

Textbook:

• V. V. Prasolov, V. M. Tikhomirov, Geometry, American Maths. Soc., 2001. (the e-book is available on Ultra (see "Reading List")

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:

Michaelmas:
• 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.

Assignments:
- There will be 4 sets of marked assignments during each term (to submit through Gradescope by noon of first Monday after weeks 3,5,7,9 and 14,16,18,20). --
- There will be also weekly unmarked sets of exercises. Please solve them timely!
- In addition, there will be additional reading material for MSc students.

Assignments:


Week #	Questions	Hints	Due date: Mon noon	Solutions	Feedback
1-2	HW 1-2	Hints 1-2	October 27	Solutions	Feedback	Isometries of Euclidean plane
3-4	HW 3-4	Hints 3-4	November 10
5-6			November 24
7-8			December 8
9-10			----

Lecture Notes: (Material 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.