MTH 783: Character varieties and 3-dimensional topology
University of Miami, Fall 2025
Instructor: Christopher Scaduto
Email: c.scaduto @ math.miami.edu
Office: Ungar 525
Office hours: 12:30-1:30 Thursday, or by appointment
Time and Location: TuTh 11AM - 12:15PM, Ungar 411
Course syllabus can be found here.
There is no single textbook for the course. References for some material covered:
Kronheimer and Mrowka, Dehn surgery, the fundamental group and SU(2): pdf
Course Schedule
| Date |
Lecture Content |
| 8/19/25 |
Introduction. Dehn Surgery on knots. |
Saveliev, Lectures on the Topology of 3-Manifolds: Lecture 2 §2.1-2.3 (see also Lecture 1 for some relevant background)
Further reading: Gordon, Dehn surgery and 3-Manifolds (2006). Many examples and exercises. (Not up to date, though.)
|
| 8/21/25 |
Wirtinger presentations. Fundamental groups of surgeries. |
|
Rolfsen, Knots and Links: The Wirtinger Presentation §3.D
|
| 8/26/25 |
SU(2) and quaternions. Representations spaces and conjugation action. |
Saveliev, Lectures on the Topology of 3-Manifolds: Lecture 13
house notes: SU(2) as unit quaternions
|
| 8/28/25 |
Stabilizer types of representations. The reducible part of the character variety. |
| house notes: Stablizers and orbits, and the reducible characters
|
| 9/2/25 |
SU(2) character varieties of knots. Example computations. The pillowcase. |
|
Kirk, SU(2) Representation Varieties of 3-manifolds, Gauge Theory Invariants, and Surgery on Knots: pdf Ch.1
|
| 9/4/25 |
Torus knots. Herald's Theorem, if time. |
|
Klassen, Representations of knot groups in SU(2): pdf IA, IB
|
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