MT A618 Course Schedule


WEEK 1: 1/13 - 1/17

Discontinuous groups acting on manifolds.
Examples of 1-,2-, and 3-manifolds.

WEEK 2: 1/20 - 1/24

No classes Monday, January 20th (MLK Day)
Last day to drop without a "W": Friday, January 24th

Introduction to "geometrization" of manifolds.
Elementary hyperbolic geometry in the plane.

WEEK 3: 1/27 - 1/31

The modular group: case study of how to find
a fundamental domain and deduce group theoretic structure

WEEK 4: 2/3 - 2/7

Dirichlet domains
Poincaré Polyhedron Theorem

WEEK 5: 2/10 - 2/14

Generalities on Coxeter diagrams and Coxeter groups
Classification of elliptic diagrams

WEEK 6: 2/17 - 2/21

Hyperbolic Coxeter diagrams, realization of Gram matrices

WEEK 7: 2/24 - 2/28

Tetrahedral groups, circle packings

WEEK 8: 3/3 - 3/7

Thurston's proof of Andreev's Theorem

WEEK 9: 3/17 - 3/21

Withdrawal deadline is Friday, March 21st

Introduction to arithmetic groups, quaternion algebras

WEEK 10: 3/24 - 3/28

Invariant trace field and quaternion algebra of a Kleinian group

WEEK 11: 3/31 - 4/4

No class Friday, April 4th because of MAA meeting

More on invariant trace field. Examples.

WEEK 12: 4/7 - 4/11

More examples. Figure 8 knot. Links from Bianchi groups.

WEEK 13: 4/14 - 4/18

No classes Friday, April 18th (Good Friday)
Trace relations, computations for polyhedral groups.

WEEK 14: 4/21 - 4/25

The Identification Theorem; Vinberg's version for Coxeter groups.

WEEK 15: 4/28 - 5/2

Friday, May 2nd is the last day of classes

Arithmetic subgroups of SO(3,1) and quaternion algebras.
Volume computations.