Math 751R: Advanced Special Topics in Topology

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Catalog Information

Title

Advanced Special Topics in Topology.

Credit Hours

3

Prerequisite

Math 551

Description

This course covers current topics of research interest.

Desired Learning Outcomes

Students should become familiar with a specific area of topology undergoing current research.


Prerequisites

Graduate level differential and algebraic topology at the level of Math 655 and 656.

Minimal learning outcomes

Minimal learning outcomes cannot be specified for a course in which topics will vary from year to year. However, regardless of the topic, successful students will know terminology, statements and approaches to problems undergoing active research, and major results in the area and techniques used to prove them. Students will demonstrate this knowledge by working suitable problems and developing their own proofs, by presenting and writing work inside and outside of class, and participating in other activities expected of more advanced graduate students.

As an example of the level and type of material covered, the following topics were required for the course when it covered hyperbolic knot theory.

  1. Link complements as 3-manifolds
    • Polyhedral decomposition associated with a link diagram
    • Alternating versus non-alternating links
    • Subdivision into tetrahedra, cusp triangulations
  2. Geometric structures on manifolds, particularly hyperbolic structures
    • Developing map, holonomy
    • Complete and incomplete structures
    • Mostow-Prasad rigidity
    • Gluing and completeness equations
    • Ideal hyperbolic tetrahedra, geometric triangulations
  3. Hyperbolic Dehn surgery
    • Hyperbolic Dehn surgery space and its shape
    • Exceptional Dehn surgeries
    • Links admitting exceptional Dehn surgeries
  4. Examples of hyperbolic links and their geometric properties
    • Volumes, embedded geodesic surfaces, etc.

Textbooks

Possible textbooks for this course include (but are not limited to):

Additional topics

Courses for which this course is prerequisite

None.