MATH4900E - Seminar - 2020/21

Course Name: 
Course Year: 

General Information


  • Prof. Yi-Jen LEE
    • Office: 412 AB1
    • Email:

Time and Venue

  • Lecture: Mondays 3:30-6:15pm via Zoom (Meeting ID and password announced via email and Blackboard)

Course Description

We will explore the interesting world of Non-Euclidean geometry, with emphasis on the most interesting case—Hyperbolic Geometry.


  • [A] Hyperbolic geometry, by James W. Anderson, Springer, 1999.
  • [BP] Lectures on Hyperbolic geometry, by Benedetti, C. Petronio, Springer-Verlag, 1992.
  • [K] Fuchsican Groups, by Svetlana Katok, Chicago Lectures in Mathematics, 1992.
  • [CFKP] Lecture notes, by J. W. Cannon, W. J. Floyd, R. Kenyon, W. R. Parry; see link below.
  • [S] Lecture notes by C. Series ; see link below.
  • [W] Lecture notes by C. Walkden; see link below.
  • [N] Visual Complex Analysis, by Needham.
  • [B] Low-dimensional geometry, by F. Bonahon.
  • [P] Notes by Pollicott; see link below
  • [KL] Hyperbolic Geometry from a Local Viewpoint, by Keen & Kakic
  • [M] The Foundations Of Geometry and the Non-Euclidean Plane, by G. Martin
  • [PM] Non-Euclidean Geometries, Edited by Preokopa & Molnar.
  • [I] Hyperbolic Geometry, by Iverson
  • [KM] The Non-Euclidean, Hyperbolic Plane, by Kelly & Matthews
  • [F] Elementary Geometry in Hyperbolic Space, by Fenchel
  • [Be] The Geometry of Discrete Groups, by Beardon

Lecture Notes

Class Notes


  • Sep 28 [Team 1: Kwok & Ma] Three types of geometries in dimension 2: Euclidean, Sperical, Hyperbolic; History of Non-Euclidean Geometry. Cf. e.g. [CFKP] Sections 1-5; [W] Section 1; [N] Chapters 1,6; [B] Chapters 1, 3; [P] Lecture 1; [KL] Chapter 1; [M]; [PM] Part I; [I] Chapter 2, Appendix; [KM] Chapters 1, 2, Appendix.
  • Oct 5 [Team 2: Cheng & Hui] Models of 2-dimensional hyperbolic space and relations among them; Hyperbolic length, lines, and distances cf. e.g. [A] ch.1, 4, 6; [BP] A.1; [CFKP] Ch.7, [W] Sections 2, 6.
  • Oct 12 [Team 3: Kung & Wong] Möbius transformations Cf. e.g. [A] Ch.2; [S], [W]
  • Oct 19 [Team 4: Kan & Lau] Hyperbolic distances and geodesics in Plane and Poincare models; Mobius transformations as isometries on the 2-d hyperbolic space; Convexity. Cf. e.g [A] Ch.3; 5.1; [S] Sect. 2, 3; [W].
  • Nov 2 [Team 1: Kwok & Ma] Hyperbolic trigonomy, polygons, Hyperbolic area; Gauss Bonnet, Hyperbolic tessellations Cf. e.g [A] Ch. 5, [W] Ch.7-8; [S] 2.2
  • Nov 9 [Team 2: Cheng & Hui] Basics of group actions and Fuchsian groups; Fundamental domains; Dirichlet polygons Cf. e.g [W] Ch.12-15; [S] Ch. 4-5; [K] Ch. 2-3; [P] (tentative)
  • Nov 16 [Team 3: Kung & Wong] Side-pairing transformations, Elliptic/parabolic cycles, Poincaré’s Theorem Cf. e.g [W] Ch.16-20; [S] Ch. 6; [K]; [P] (tentative)
  • Nov 23 [Team 4: Kan & Lau] Gluing constructions; hyperbolic surfaces (possibly with cusps); Euler characteristics; signature of Fuchsian group; covering spaces; uniformization Cf. e.g. [W] Ch. 21-22 (including Exercise 21.2; 21.3 (i)); [S] Ch.7; [K] Ch. 4; [Be] Ch.6.

Assessment Scheme

Oral presentations  100%

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Last updated: November 16, 2020 17:16:52