Geometric 3-Manifolds and the Poincare's Fundamental Polyhedron Theorem -- Max Lipyanskiy, April 16, 2002

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This talk is intended as a general introduction to some of the concepts and 
theorems of 3-manifold topology. I will focus on several theorems from the 
following selection of topics: <BR><BR>

1) Topology of closed manifolds in dimension2 and 3 - Cutting along circles and spheres (Thm of Kneser/Milnor), JSJ 
Decomposition <BR>
2) Hyperbolic geometry - Basic models, geodesics, isometries, curvature calculations <BR>
3) Gauss-Bonnet Theorem - Proof for the case of constant curvature <BR>
4) Geometric Manifolds - Basic geometries, Mostow Rigidity, discussion of invariants, Thurston's Geometrization Conjecture <BR>
5) Discrete Subgroups of PSL(2,C) - Descriptions of discreteness, fundamental polyhedra, Poincare's theorem <BR>
6) My work during 2001-2002 - Seven exceptional regions in "Homotopy Hyperbolic Manifolds are Hyperbolic", a 
rigorous computer application of Poincare's Theorem