Math 125: Geometry of Discrete Groups
Summer 2015
Course Info:
- Lectures: Monday, Wednesday, Friday, block 10 (10:00 a.m. - 11:05 a.m.)
- x-period: Thursday 12:00 noon - 12:50 p.m.
- Dates: 26 June 2015 - 26 August 2015
- Room: 004 Kemeny Hall
- Instructor: John Voight
- Office: 341 Kemeny Hall
- E-mail: jvoight@gmail.com
- Office hours: Wednesday 1:00-2:30 p.m., Thursday 1:00-2:00 p.m., or by appointment
- Course Web Page: http://www.math.dartmouth.edu/~m125x15/
- Prerequisites: None.
- Required Texts: None (course notes).
- Grading: Grade will be based on participation and homework.
Syllabus:
[PDF] Syllabus
This course will provide an introduction to the geometry of discrete groups. Specifically, we will study the action of discrete subgroups (lattices) inside groups of isometries of hyperbolic spaces, especially dimensions 2 and 3. An interesting class of such groups arises from arithmetic constructions, and the corresponding quotients give rise to orbifolds and manifolds of interest in number theory, geometry, spectral theory, combinatorics, and topology.
Homework:
The homework assignments will be assigned on a weekly basis and will be posted below.
Cooperation on homework is permitted (and encouraged), but if you work together, do not take any paper away with you--in other words, you can share your thoughts (say on a blackboard), but you have to walk away with only your understanding. In particular, you must write the solution up on your own. Please acknowledge any cooperative work at the end of each assignment.
Plagiarism, collusion, or other violations of the Academic Honor Principle, after consultation, will be referred to the The Committee on Standards.
[PDF] Homework Submission Guidelines
Please sign up for problems on the Homework cribsheet.
Bibliography (as of 1 July 2015)
| 1 | 26 Jun | (F) | Introduction, geodesic spaces (27.1) | Chapter 27: Hyperbolic geometry |
| - | 27 Jun | (S) | No class, 8:00 a.m. on a Saturday, srsly? | |
| 2 | 29 Jun | (M) | Upper half-plane, isometries, geodesics (27.2-27.4) | |
| 3 | 1 Jul | (W) | Area, unit disc, Lorentz model (27.5-27.6) | HW 1 |
| 4 | 2 Jul | (R) | Hyperbolic space (27.7) | |
| - | 3 Jul | (F) | No class, College holiday | |
| - | 6 Jul | (M) | No class, JV in Oberwolfach | |
| - | 8 Jul | (W) | No class, JV in Oberwolfach | |
| - | 10 Jul | (F) | No class, JV in Oberwolfach | |
| 5 | 13 Jul | (M) | Topological group actions (28.1-28.2) | Chapter 28: Discrete group actions |
| 6 | 15 Jul | (W) | Hausdorff, proper group actions (28.3) | HW 2 |
| 7 | 16 Jul | (R) | Symmetric space model, Fuchsian groups (28.4-28.5) | |
| 8 | 17 Jul | (F) | SL2(Z) (29.1) | |
| 9 | 20 Jul | (M) | Binary quadratic forms (29.2) | Chapter 29: Classical modular group (fixed 29.3 on Jul 23) |
| 10 | 22 Jul | (W) | Functions on lattices, complex tori (29.3) | HW 3 |
| 11 | 23 Jul | (R) | Modular forms (29.4) | |
| 12 | 24 Jul | (F) | Kleinian groups, SL2(Z[i]) | |
| - | 27 Jul | (M) | No class, grrr, JV on tarmac | |
| 13 | 29 Jul | (W) | Hyperbolic volumes (30.6) | |
| 14 | 30 Jul | (R) | Dirichlet domains for Fuchsian groups (31.1) | |
| 15 | 31 Jul | (F) | Ford domains (31.2) | |
| 16 | 3 Aug | (M) | Generators (31.4) | Chapter 31: Fundamental domains (fixed typos on 5 Aug) |
| 17 | 5 Aug | (W) | Side pairings (31.3) | HW 4 |
| 18 | 6 Aug | (R) | Examples | |
| 19 | 7 Aug | (F) | Poincare's theorem (31.5) | |
| - | 10 Aug | (M) | No class, JV at AIM | |
| - | 12 Aug | (W) | No class, JV at AIM | |
| - | 14 Aug | (F) | No class, JV at AIM | |
| 20 | 17 Aug | (M) | Motivation, quaternion algebras | Chapter 1: Introduction, Chapter 2: Beginnings |
| 21 | 19 Aug | (W) | Splitting, involutions | Chapter 3: Involutions, Chapter 8: Orders, Chapter 11: Quaternion algebras over local fields |
| 22 | 20 Aug | (R) | Orders, discriminants | HW 5 |
| 23 | 21 Aug | (F) | Quaternionic groups: discriminant 6 (32.1) | |
| 24 | 24 Aug | (M) | Quaternionic groups: discriminant 6 | |
| 25 | 26 Aug | (W) | Quaternionic groups | |
| 26 | 27 Aug | (R) | Wrapup |