(John Parker)
DescriptionThe geometrisation conjecture was posed in 1982 by William Thurston. It says that any 3-manifold may be decomposed canonically into pieces and that each of these pieces carries one of eight geometries. Moreover, which geometry this piece carries is determined by topological information. A special case of this is the Poincaré conjecture which says that every simply connected, closed 3-manifold carries the geometry of the 3-sphere. Thurston's approach revolutionised the theory of 3-manifolds. In 2003 Grigori Perelman announced the sketch proof of geometrisation conjecture, including the Poincaré conjecture.In this project we will discuss what is meant by the word geometry in the above paragraph. We will then discuss the eight geometries in detail. There will then be scope for each student on the project to discuss one of these eight geometries in detail. The project will be supervised by John Parker.
ResourcesIn the first term we will work through the survey article The geometries of 3-manifolds by Peter Scott, published in Bulletin of the London Mathematical Society 15 (1983), 401--487.For some more details see wikipedia. Youtube has a lecture by Thurston on the geometrization conjecture from 2001. |
email: J R Parker