Difference between revisions of "Math 751R: Advanced Special Topics in Topology"
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#*Subdivision into tetrahedra, cusp triangulations | #*Subdivision into tetrahedra, cusp triangulations | ||
#Geometric structures on manifolds, particularly hyperbolic structures | #Geometric structures on manifolds, particularly hyperbolic structures | ||
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#*Complete and incomplete structures, Mostow-Prasad rigidity | #*Complete and incomplete structures, Mostow-Prasad rigidity | ||
#*Gluing and completeness equations, ideal hyperbolic tetrahedra | #*Gluing and completeness equations, ideal hyperbolic tetrahedra | ||
Revision as of 09:29, 16 July 2010
Contents
Catalog Information
Title
Advanced Special Topics in Topology.
Credit Hours
3
Prerequisite
Math 655 and 656, or instructor's consent.
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, students will be expected to know terminology, statements and approaches to motivating 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 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.
- Link complements as 3-manifolds
- Definition of a link complement
- Polyhedral decomposition associated with a link diagram
- Alternating versus non-alternating links
- Subdivision into tetrahedra, cusp triangulations
- Geometric structures on manifolds, particularly hyperbolic structures
- developing map, holonomy
- Complete and incomplete structures, Mostow-Prasad rigidity
- Gluing and completeness equations, ideal hyperbolic tetrahedra
- Hyperbolic Dehn surgery, hyperbolic Dehn surgery space
- Examples of hyperbolic links and their geometric properties
- Volumes, embedded geodesic surfaces, etc.
Additional topics
Courses for which this course is prerequisite
None.
