Alan Reid, Professor of Mathematics at the University of Texas

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Spring 2018
Math 542: Topics in Advanced Topology–Arithmetic of Hyperbolic Manifolds

This class will study arithmetic aspects of hyperbolic manifolds. This will largely be confined to finite volume hyperbolic manifolds in dimensions 2 and 3, but if time allows a discussion of higher dimensions will be given. Associated to any finite volume hyperbolic 2 or 3-manifold M is a pair (AM, kM) which consists of a quaternion algebra AM over a field kM which is an invariant of the commensurability class. In particular, in dimension 3, kM is a number field, and algebraic and arithmetic invariants of this field and algebra are powerful tools in the study hyperbolic 3-manifolds.

The main aim of this course is to develop this. In the special setting of arithmetic hyperbolic 3-manifolds (and 2-manifolds) this pair underpins much of the geometric, topological and spectral properties of the manifold.

There is no set text book. Occasional HW sheets will be given out for self-study and group discussion. These are not to be handed in. There will be no exams.

Pre-requisites: The graduate topology and algebra courses, some number theory would be helpful, but not absolutely necessary.

Class meets: 1:00PM - 1:50PM MWF

Room: HBH 423

First day handout (pdf)

Homework

1. pdf

2. pdf

3. pdf

4. pdf