** **## Paper Guidelines

###
** Final paper **

- Your final paper should present, in an expository way, a topic related to but
not covered in the course.
- The final paper should be about 10-20 pages.
- Papers typeset in TeX are weclome!
- A short paper showing you have understood a topic thoroughly is
the best.
- Try to make your treatment as concrete as possible -- include examples
as well as theory. The more focused the better.
- An outline of your final paper, with references, is due on
Monday, 24 November.
- The final paper is due on the last day of the course,
15 December.
- You will present your final project to the class during the last
2 meetings of the course, 7 and 15 December.

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** Some Possible Topics **

- The Jordan Curve Theorem and the Schoenflies Theorem
- Immersed loops in the plane (Whitney)
- Embedded / immersed surfaces in space
- Hyperbolic groups (Gromov)
- The boundary of a group
- Unsolvable problems in group theory
- The modular group SL
_{ 2}(Z) and the groups SL_{ 2}(Z/n)
- Hyperbolic geometry of surfaces
- Circle bundles over surfaces and the Hopf invariant
- Simple closed curves on surfaces
- The figure eight knot complement
- Fibered knots
- Vasilliev invariants
- Hyperbolic volume as a knot invariant
- Wild knots, Antoines' necklace and the Alexander horned sphere
- Dehn's Lemma and/or the Loop Theorem

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** Midterm paper **

- The midterm paper is a warmup to the final.
It should present a concise exposition of a topic that has been
discussed or touched on in class.
- The midterm paper should be about 5-10 pages.
- The midterm paper will be due on 3 November 2003.

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