Wednesday 4:00-5:00
Room: 4416
This is an informal seminar studying the above book by William Thurston published by Princeton University Press in 1997, meeting once a week during the fall semester of 2005. We hope to continue these seminars in the winter of 2005 and the fall of 2006 as well. I'll basically give all the talks. The course should be elementary and undergraduates are welcomed to come to the meetings.
The book is now an elementary one covering the introductory part of the original note "Geometry and Topology of 3-manifolds".
Basic plans:
Introduction to hyperbolic spaces: using Ratcliffe and Beardon
Then we will basically follow Thurston's book.
Topics: Hyperbolic spaces, isometry groups
Chapter 1:
What is a manifold? (This is an introduction)
Chapter 2:
Hyperbolic geometry and its friends (This explains hyperbolic space,
isometry,
computations on hyperbolic spaces)
Chapter 3:
Geometric manifolds (Defines geometric structures, developing
maps,
8 model geometries in 3-dimensional spaces)
Chapter 4:
Structures of Discrete groups (Bieberbach theorem, Euclidean and
elliptic
manifolds, thick and thin decompositions, Teichmuller
space)
If we are very hard working, we may start working on "The geometry and
topology of three-manifolds" (http://www.msri.org/publications/books/gt3m/)
References:
Thurston: Thee-dimensional geometry and topology, Princeton University
Press
Ratcliffe: Foundations of hyperbolic manifolds, Spinger Verlag
Beardon: The geometry of discrete groups, Springer Verlag
Wolf: Spaces of constant curvature, 5th edition, Publish or Perish
Since there are many interesting related websites, I will add them here:
Geometry Center: http://www.geom.uiuc.edu
There is a video introducing the hyperbolic three-space: http://www.geom.uiuc.edu/graphics/pix/Video_Productions/Not_Knot/
There are many other resources here: programs, introductory articles,
and so on.
Geometry games by J. Weeks: http://www.geometrygames.org/
In particular, Snappea is a program to compute hyperbolic
structures on 3-manifolds.
Experimental Geometry Lab (University of Maryland):http://egl.math.umd.edu/
There are nice pictures and programs.
Computop.org softwares: http://www.its.caltech.edu/~dunfield/computop/index.html
Many research level softwares for 3-dimensional topology
Knitting and topology: http://www.theiff.org/oexhibits/oe1e.html,
http://cerebro.cs.xu.edu/~smbelcas/math-knit.html